The multiculturality of mathematics
Terezinha Nunes
Department of Educational Studies
University of Oxford
Most people would readily agree that mathematics
learning involves using logic and most mathematics educators
today would accept that mathematics is a cultural product – so,
learning mathematics is learning to master a cultural invention.
Both of these are appealing ideas but they can be seen as conflicting:
logical principles should be universal and cultural inventions
are not. In this paper, I will try to show how these ideas
can be made compatible by working with the notion of reasoning
systems. I will then explore the notion of reasoning systems
in three contexts of mathematics learning: learning to count,
using counting to solve different arithmetic problems, and
translating (or rather failing to translate) between different
representations for mathematics.
In exploring these examples, I will be working
with some assumptions, which I want to make explicit from the
start. First, I will assume, following the Piagetian tradition,
that children's logical reasoning has its origins in their
actions. This assumption is used in the design of the studies
I will describe, in which children are asked to solve problems
and the way they organise their actions is taken as an indication
of their logic. Second, I will assume that the mathematical
representations that they use are culturally provided, and
have to be integrated into their reasoning system in order
to be used. It does not matter for this discussion how the
representations are learned; the argument is only that they
are used well only when they are assimilated into a reasoning
schema. Finally, I will be proposing that, once these representations
are assimilated into a reasoning schema they affect the way
it works, usually by increasing its power but also by structuring
the way in which we deal with information. This means that,
once some representations become part of the system, they direct
our interpretation of new information and our problem solving
efforts in specific ways. When there are alternative representations
for the same situations, we may not easily move between them – but
it would be to our advantage to do so.
What are Reasoning Systems?
Systems theory was applied to reasoning both
by Piaget and the Russian developmental psychologists in the
first half of the twentieth century. Piaget and the Russian
developmental psychologists were attempting to solve the same
problem and envisaged systems theory as the solution. The problem
they were trying to solve was the mind-body problem. The problem
is easily understood by considering the contrast between biological
and higher mental functions.
Biological functions are typically carried
out by specialised organs. For example, digestion is carried
out by the digestive system; breathing by the respiratory system.
Biological functions involve a constant task performed by the
same mechanisms leading to an invariant result. If we consider
breathing as an example, the task is to bring oxygen to the
cells in the body. This is accomplished by an invariant mechanism:
oxygen is received by the blood cells and transported to all
the cells in the body. The invariant result is that the cells
receive oxygen.
In contrast, higher mental functions are not
carried out by a specialised organ but through the co-ordination
of different actions. They are carried out by functional systems.
According to Luria's definition, in functional systems “a constant
task [is] performed by variable mechanisms bringing the process
to a constant result” (Luria, 1973, p. 28). I will take one
of Luria's examples to make this point. The first one is ‘remembering'. It
is easy to be misled into thinking that we have a specialised
organ for remembering: the brain. But Luria points out that
remembering involves functional systems rather than a single
biological unit. Imagine it is your partner's birthday and
you want to remember to buy some flowers before going home.
Your task is to remember to buy flowers. You can accomplish
this through a variety of means. You can simply repeat this
to yourself many times until you think it is now impossible
for you to forget. You may tie a knot around your finger: as
you don't normally have a string around your finger, this will
remind you to buy the flowers. You might write it down to help
you remember – on your palm, where it will be very visible.
Or on a yellow sticker, for example, and paste it on your wallet.
Or you may type it into your electronic diary and set an alarm
to go off just before you leave your office. These variable
mechanisms can be used with the same end: to recover the information.
No single biological unit can account for all the different
mechanisms you may call upon when trying to remember something,
and most of us do not rely on “natural” memory to organise
our lives: we have diaries and write into them what we are
supposed to do when and we use Powerpoint when giving seminars
to help us remember what we wanted to say next. Human memory
is limited but humans don't depend on what they can remember
without help: our memory system is open in the sense that it
can work with external aids, and we are empowered by learning
to use external memory mechanisms that, in conjunction with
our natural memory, allow us to remember better. Many cultural
aids work in ways that help us surpass the limits of our memory.
The second example I want to explore relates
to external mechanisms that help us surpass the limits of our
perception of time. A thought experiment might help understand
what I mean by this. Suppose I ask someone: when are we meeting
tomorrow? If this is a culture without clocks, what type of
answer could I have? How does the answer change when the question
is asked in a culture that has clocks? In a culture without
means of measuring time the answer must differ because it will
be based on perceptions related to time whereas in a culture
with clocks we will use the measurement of time to provide
our answer. In a culture without clocks, our idea of how long
is the day and how long is the night might depend on whether
it is winter or summer; in a culture with clocks, we define
a day as lasting a certain number of hours even if it gets
dark earlier in the winter and later in the summer. So our
conception of time becomes independent of our perceptions – even
if we cannot perceive the difference between 10 o'clock and
5 past 10, we can still look at the clock and tell the other
person: “you are late”. Our conceptions of time in cultures
where we rely on clocks are shaped by the measurement of time – we
think of the day as having 24 hours, the hours as having 60
minutes, the minutes as having 60 seconds as if this is what
time is. But we could think of the same cycle has having 86,400
seconds, organized into minutes of 100 seconds, and might say
nothing about hours. In this case, instead of saying “I will
see you at 10 o'clock tomorrow”, we would say “I will see you
at 360 minutes tomorrow”. Having a way of measuring time,
we incorporate it into our reasoning system about time.
Reasoning systems are open systems: they allow
for the incorporation of tools that become an integral part
of the system. Vygotsky suggested that what is most human about
humans is this principle of construction of functional systems
that allow activities to be mediated by tools. He termed this ‘the
extra-cortical organisation of complex mental functions' to
stress that these functional systems cannot be reduced to the
brain.
Even the most elementary mathematical activities
are carried out by functional systems. Solving the simplest
addition problem, for example, involves a reasoning system.
Paraphrasing Luria: we have no specialised organ for addition.
If asked to solve the problem ‘Mary had five sweets and her
Grandmother gave her three more; how many does she have now?',
a pupil can find the answer through a variety of mechanisms.
The pupil can put out five fingers, then another three, and
count them all. The pupil can put out just three fingers and
count on from five. The pupil can recall an addition fact,
5+3, and use no fingers. In this were a large number, the pupil
might decide to use a calculator. These are variable mechanisms
that bring the invariant result of finding the answer to addition
problems.
For educators, one of the most significant
features of higher mental functions is that they are open systems:
the variable mechanisms – which are often created through the
incorporation of tools - can be replaced by taking into the
system something new from the environment. When a mechanism
is replaced with something new, the system changes. To use
Piaget's terminology, when a child assimilates something new
into his or her reasoning system, the system accommodates to
a new way of functioning.
Many of the changes that we introduce into
children's mathematical reasoning systems in school are representational:
we teach children to represent things in new ways which empower
them. These new ways of representing, in turn, have an impact
on their reasoning system, and open to them new possibilities
of reasoning. This is not, of course, all that happens in mathematics
learning, but it is a very important part of what happens in
mathematics learning. The idea of reasoning systems will be
illustrated here with three examples: understanding counting,
using counting to solve different types of problems, and different
systems for handling addition and subtraction and intensive
quantities.
Counting
In order to learn to count appropriately,
children must be able to follow three principles: to establish
a one-to-one correspondence between number labels and objects,
to have a different number label for each object to be counted
(otherwise the one-to-one correspondence principle would be
violated) and to keep the number labels in a fixed order (otherwise
counting the same number of objects might end up on a different
number label on different occasions). This means that counting
1,000 objects would require that children learn 1,000 different
words in a fixed order – a major task for our poor memory.
Over the course of history, cultures have solved this problem
by using a base-system for counting. With a base system, we
do not have to remember the counting words, we learn how the
counting system works, and can generate new counting words
which we have not heard before.
Base systems could be used simply as a series
of words that are easy to generate. However, they represent
a new idea, which goes beyond one-to-one correspondence: they
represent the additive composition of number. Children may,
at first, use a counting system with a base without grasping
this principle, and only later grasp the idea of additive composition
that is part of a base system. We have for many years used
the Shop Task as a simple experiment that shows whether the
children understand additive composition. In this task, children
are asked to pretend that they are buying things from the experimenter
and offered coins that they can use to buy these things. Under
the one-to-one correspondence condition, the children are given
many 1p coins and ask to pay, for example, 13p, 17p, 23p and
25p for different toys. All they need to do is to count the
coins one by one until they reach the desired number. Our studies
have shown that children in their first year of school in the
UK (mean age in our last study 5 years 10 months, N=112) have
no difficulty in counting the desired amounts of money in this
one-to-one correspondence condition, and reach almost 100%
correct responses. Under the additive composition condition,
the children are given coins of different values: to pay 13p
and 17p, they are given one 10p coin and 9 1p coins, and to
pay 23p and 25p they are given one 20p coin and 7 1p coins.
Their performance in this second task is quite different: 51%
of the children give no correct response (in six items), counting
the 10p and 20p coins as ones, even though they know the value
of the coins. They do not use the coins of different values
to compose quantities even though they can count up to those
quantities when they are given only 1p coins. We conclude that
these children are using a base system as if it were a simple
string, which only required one-to-one correspondence principles.
Over time, and also with the support of teaching, the children's
reasoning accommodates to the additive composition principle
embedded in the system, and through this the children gain
more power in the use of the counting system.
Much research has shown that the regularity
of the counting system affects how easily the children learn
to count (e.g. Miller & Stigler, 1987; Nunes & Bryant,
1996) and also how easily they grasp the idea of additive composition
(e.g., Nunes & Bryant, 1996; Miura et all, 1988). This
research can be summarised by examining one of our comparisons
betweenTaipei and>Oxford children. The Chinese counting system
is quite regular, and the words for 11, 12 etc are the equivalent
to ten-one, ten-two etc. The words for 20, 30 and the other
decades are literally translated as two-ten, three-ten etc.
21, 22 etc literally translated as two-ten-one, two-ten-two
etc. This regular system, which represents the idea of additive
composition so clearly, contrasts with English, where the count
words in the teens are irregular. Note, however, that neither
system gives any clues to the idea of additive composition
when the children have to pay 7p using a combination of five
plus two ones. So, if the Taiwanese children perform better
when producing these combinations of coins, this shows that
they are not merely matching words to coins but that they have
understood the principle of additive composition. We gave the
5-year-old children the Shop Task using the relevant currency
in each country and varying the denominations of coins: some
trials used one ones, others used combinations of five and
ones, and other used combinations of ten and ones. Within this
range, theTaipei and>Oxford children did not differ when paying
for toys in the Shop Task when the coins were all ones. However,
they differed significantly when they had to use the additive
composition principle. The>Taipei children made very few errors
(8% only) in the combinations of ten and ones, where their
system can help them with the idea of additive composition
very directly, and a few more (20%) when they had to count
combinations of five and ones, where they have no direct help
from the system. In both cases, they performed significantly
better than the >Oxford children, who gave wrong answers to
56% of the trials that used combinations of ten and ones and
to 46% of the trials that used combinations of five and ones.
Our conclusion is that the reasoning system
that children form in order to count depends on logic – the
correspondence principles mentioned earlier on and the additive
composition principle, when using a base system – and also
on the culturally devised tool for counting, a system of number
labels. By learning the system of number labels in English
or in Chinese, children overcome the limits of their natural
memory, because they can generate the number labels by using
the logic of the system. If this logic is more transparent
in the system, they can understand it more quickly.
Using counting to solve different types of
problems
From the preceding examples, it could appear
that counting is related to addition in a straightforward way – that
knowing how to count and how to add are almost the same thing
and that there is nothing more to counting than adding. In
order to make the point that counting is a tool in arithmetic
in general, not just a basis for addition, it suffices to consider
a few examples where children's logic guides the way they organise
materials in order to count and to find the answer to different
types of problems. In these examples, the children's logic
guides the way they manipulate objects and what they count,
once they have represented the problem.
The first example is a contrast between two
subtraction problems. Both are change problems but in one the
missing value is the end result and in the other the missing
value is the change. A change problem with result unknown would
be: a boy had 6 marbles; he put them in his pocket and went
out; he had a hole in his pocket and 4 marbles fell out; how
many did he have when he got home? The same problem with the
change unknown: a boy had 6 marbles; he put them in his pocket
and went out; he had a hole in his pocket and some fell out;
when he got home, he had 4 marbles left; how many fell out?
In both problems, the child can use the materials to represent
the total, represent the change, and the result. In the first
problem, the child removes 4 marbles and counts the ones left.
In the second problem, the child counts 4, which are the end
state, and has to conclude that the others are the marbles
left in the boy's pocket. Although the problems seem so similar
when the child acts the problem out, the missing end state
problem led to 75% correct responses in our sample of children
in their first year of school whereas the missing transformation
problem led to 35% correct responses. The child has the same
tools but different logical moves are required.
The second example is the use of counting
to solve multiplication problems. We gave children the following
multiplication problem: in each house in this street (four
houses are drawn) live three dogs. How many dogs live in this
street? In order to solve this problem, children count by pointing
to the houses – not in one-to-one correspondence but in one-to-many
correspondence because they point and count three times as
they point to each house. Children's performance in these problems
is surprisingly good: 58% of the children's responses (to 5
problems) were correct when they were in their first year of
school (5y10m old, N=112).
These different ways in which children use
counting illustrates how their logic guides the way they count
in order for them to solve problems. Knowing how to count is
a tool, necessary but not sufficient, to solve problems, much
like having a watch is necessary but not sufficient for children
to be able to tell the time.
Translating between tools: addition and subtraction
calculation ability
Children first solve addition problems using
their fingers to represent the objects in the problem. The
principle used by the pupils' reasoning system in this representation
is one-to-one correspondence: one finger represents one sweet;
one counting word is tagged to one finger; the last counting
word indicates the number of sweets. There is an important
logical move underlying this way of solving problems: if children
understand that they can count fingers to solve problems about
sweets, they understand that the referent (finger, sweet) does
not affect the result of an addition operation. In other words,
1 + 1 = 2 regardless of what the numbers represent.
Later, children are able to “count on”: when
solving the problem “A girl had 5 sweets; her Granny gave her
3; how many does she have now?”, they replace the use of five
fingers with the word ‘five' by itself. This change seems to
be a representational change: instead of one finger for each
sweet, one word – five – represents all five sweets at the
same time. This small change has a huge impact on the reasoning
system: whereas children have a limited number of fingers,
number words continue indefinitely on. A system with fixed
limits becomes much more powerful because its limits are removed
by a change in tools.
There are some indications in our research
that this representational change is related to children's
understanding of additive composition. A study carried out
by Katerina Kornilaki gives strong support to this idea. She
gave children different types of tasks, which involved counting
with different levels of representation of one of the addends – the
other addend was always visibly represented. In the first type
of task, the children were asked to say what where the totals
of two sets which were represented visually and remained represented
visually the whole time. So they could simply count all the
visible objects. In the second type of task, adopted from Steffe
and his colleagues (1982), the children were asked the sum
of two sets, but one of them was counted and then hidden – for
example, the children were told that a girl had six coins,
saw the coins, and these were then put into a wallet. Then
the girl was given three coins, which remained visible; the
children were asked how many coins the girl had now. Hidden
addend tasks, as Steffe found, could be solved either by count
all or count on procedures. When counting all, the children
pointed to the wallet inside which there were coins, and counted
up to the six, and then went on to count the three coins outside.
When counting on, the children simply said “six” and counted
on. The third type of task was our Shop Task. The difference
between the tasks is in the representation of the first addend:
it is either completely visible, or visible at first but then
hidden, or encoded in a coin of a value larger than 1. The
tasks clearly vary in level of difficulty: the first being
is easiest and the third, the Shop Task, the most difficult.
An analysis of the children's performance across tasks showed
that only children who were able to count on in the hidden-addend
task succeeded in the Shop Task – but not all of them did so:
counting on was a necessary but not sufficient move to understand
additive composition. A change in the representational tools – from
the expanded representation in one-to-one fashion to the compressed
representation of the Shop Task has an impact on the children's
performance because of the extra logical move, additive composition,
required to succeed in the Shop Task.
The understanding of additive composition
is a powerful idea and seems to be the basis for addition and
subtraction of larger numbers in oral arithmetic. The reasoning
system of oral arithmetic can develop in the absence of knowledge
of written arithmetic. In >Brazil, these two systems exist
as independent cultural practices: oral arithmetic is used
in street markets and the informal economy in general and written
arithmetic is used in schools. The contrast between the two
systems illustrates how the same logical principles create
different reasoning systems when the cultural tools used by
the person are different. Our previous work in >Brazil has
shown that young people and adults with little schooling may
have very good calculation ability in the oral mode and poor
calculation ability in the written mode. This difference is
not explained by differences in logical principles used to
calculate using oral arithmetic in comparison to written arithmetic:
it is explained by the differences in the cultural tools being
used. The analysis of children's calculation procedures in
oral and written arithmetic shows that both rely on the logic
of the additive composition of numbers and the associative
property of addition. If 17 is the same as 10 plus 7, and 13
is the same as 10 plus 3 (additive composition), the sum of
17 plus 13 can be found by adding the tens, adding the ones,
and then putting it all together. These are the principles
used when children carry out additions and subtractions orally
and also the principles used in the written algorithm. However,
because the tool for representation changes – oral or written
numbers – the reasoning systems used in calculation are different.
They can be integrated but they can also exist in isolation
from each other. Our work shows that children who are expert
at oral arithmetic may nevertheless find it difficult to use
the same reasoning principles to calculate with the written
system (a discussion of this research can be found in Nunes,
Schliemann and Carraher, 1993) and vice versa – many people
can use the written system to calculate but not the oral system.
Translation between oral and written calculation systems is
not simple. However, it is possible to envisage a system that
coordinates both of these, which would be more powerful and
flexible than either one.
Translating between tools: ratio and fractional
language in intensive quantities
The final example I will use here is the
representation of intensive quantities and how it affects children's
understanding of intensive quantities problems. Intensive quantities
are quantities defined by the ratio between two other quantities.
For example, the taste of orange juice is dependent on the
ratio between orange concentrate and water. The probability
of an event occurring (for example, to draw a blue marble from
a bag) is dependent of the ratio between the favourable and
non-favourable cases (the marbles of other colours). Many (though
not all) intensive quantities can be represented numerically
by ratios or fractions. The mixture of orange juice can be
expressed as one cup of concentrate for two cups of water or
1/3 concentrate. The probability of drawing a blue marble can
be described as one blue marble for two white marbles or as
1/3. Both forms of representation involve proportional reasoning
when making comparisons – for example, between the taste of
orange juice of two mixtures or the probability of drawing
a blue marble from two bags with a mixture of white and blue
marbles.
Despina Desli compared children's ability
to solve some intensive quantity problems when the information
was presented to them in ratio language or in fraction language.
The children were given problems like this one: A girl made
three cups of orange juice by mixing one cup of concentrate
with two cups of water. The juice tasted perfect so the next
day, when she was going to make juice for a party, she wanted
it to taste exactly the same. She had to make 18 cups of juice.
How many cups of concentrate and how many cups of water should
she use? This problem is couched in ratio language; half
of the children had the problem presented to them in this way.
The other half had the problem presented in fraction language:
A girl made three cups of orange juice by using a mixture of
1/3 concentrate with 2/3 water. The juice tasted perfect so
the next day, when she was going to make juice for a party,
she wanted it to taste exactly the same. She had to make 18
cups of juice. How many cups of concentrate and how many cups
of water should she use? Half of the children in each group
had manipulatives, which they could use to help them solve
the problem – little cups coloured in orange and white. The
other half of the children only had paper and pencil. The children
were assigned randomly to these conditions.
All the children in the study, aged 8 to
10, had been taught in school about the fractions used in the
problems. The 8- and 9-year old children who were presented
with the problems in ratio language performed significantly
better than those who were presented with the problem in fractions
language; the difference was not significant for the 10-year-olds.
Because the children had been randomly assigned to these testing
conditions, it is reasonable to assume that the language differences
explain the differences in performance. It is possible that
children can start to develop their understanding of proportionality
by using a ratio representation without being able to connect
it to a fractional representation – and only later achieve
this coordination.
An analysis of children's strategies suggested
that this is the case. When the problems were presented in
ratio language, 30% of the children were able to use correspondence
reasoning, and replicate the correspondences until they reached
the total amount. For example, one cup concentrate with two
cups water makes 3 cups; 2 cups concentrate with 4 cups water
would make 6 cups – and so on, until they had the desired 18
cups total. The logic of correspondences develops early and
children can start to solve proportions problems by using it.
When presented with the information in fractions language,
it is not so easy to use this schema. Because children do not
find it easy to translate back and forth between the two ways
of representing problems, even though they could have used
correspondences to solve the problem, very few (4%) of those
who had the problem presented in fractions language thought
of this solution.
Conclusions
These examples were used to show how the
idea of reasoning systems, which are open and can accomplish
the same aim through different mechanisms, can coordinate the
use of logic with different tools that are culturally developed.
The tools can both empower the user, by helping overcome the
limits of memory and perception, for example, and also will
influence how the user conceptualises mathematical problems.
Because different representational tools can highlight different
aspects of the same situation, users who can coordinate them
are more flexible. However, this coordination might not take
place easily and spontaneously: one of the aims of mathematics
teaching should be to help users coordinate systems that seem
to work independently but relate to the same logic.
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