Children’s perception of the affordances of the
mathematical tools
Yasmine Abtahi
To cite this version:
Yasmine Abtahi. Children’s perception of the affordances of the mathematical tools. Konrad
Krainer; Naďa Vondrová. CERME 9 - Ninth Congress of the European Society for Research
in Mathematics Education, Feb 2015, Prague, Czech Republic. pp.2440-2445, Proceedings of
the Ninth Congress of the European Society for Research in Mathematics Education. <hal01289326>
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Children’s perception of the affordances
of the mathematical tools
Yasmine Abtahi
University of Ottawa, Ottawa, Canada
Vygotsky’s theory proposes a sign/tool-mediated view
of learning. Tools and signs are inseparable parts of
teaching and learning of Mathematics. Vygotsky’s theory provokes questions: how are tools perceived, how
are signs tied to the use of tools and consequently how
are tools being used in the mathematics classroom. In
this paper, I look at Vygotsky’s perspectives on the perception of the tools, through the lens of Gibson’s view of
affordances. I analyse three children’s interactions with
the mathematical tools, as they gradually begin to tie
signs to them, while working on addition of fractions
problems.
Keywords: Perception, affordances, tools, fractions,
meaning.
INTRODUCTION
Fractions are one of the most challenging concepts
to teach and learn in elementary-school mathematics (Steffe & Olive, 2010). Lamon (2007) noted that
fractions are one of the topics in elementary-school
mathematics that are among ‘the most difficult to
teach, the most mathematically complex, the most
cognitively challenging, and the most essential to
success in higher mathematics and science’ (p. 23).
One way to assist children in learning fractions is to
employ different mathematical tools in the classroom
(Cramer & Henry, 2002; Misquitta, 2011; Mendiburo,
2011; DeCastro, 2008; Mills, 2011; Cramer, Post, & del
Mas, 2002). Extending Swan and Marshall’s (2010)
definition, mathematical tools are any tool-like objects
that can be handled by an individual during which
mathematical thinking is fostered. Objects – any foci
of attention (Engestrom, 2009) – are inseparable parts
of any mathematics classroom. They include tools
such as an abacus, symbols such as x², and graphs.
The use of tool-like objects refers to Marx’s view of
the use of working tools; where man uses the phys-
CERME9 (2015) – TWG16
ical and mechanical properties of objects to reach
his goals. Hence, an abacus is a tool-like object and x²
is not, because children are able to use the physical
and mechanical properties of an abacus to achieve a
mathematical goal.
Although the use of mathematical tools is conceptualised as being useful in the learning of fractions,
children encounter difficulties in grasping the relationship between mathematical tools and the mathematical meanings that they are intended to represent (Norman, 1993; McNeil & Uttal, 2009; Rabardel &
Samurçay, 2001). The process within which the child
grasps the interrelationship between mathematical
tools and the meaning of a mathematical concept is a
highly complex one.
McNeil and Uttal (1997) explained that any mathematical tool “can be thought of in two different ways: (a) as
an object in its own right and (b) as a representation
of something else” (p. 43). For example, if the relationship between the sizes of the pieces in fraction circles
and the concept of the additions of fractions is not
clear to a child, then he/she needs to learn not only
the mathematical concept, but also the functionality
of the fraction circles as a system and its relationship
with the mathematical concept; in other words, he/she
needs to learn two separate systems and the relationship between them.
In this paper, I look at the interrelationship between
mathematical tools and the mathematical meanings
represented by the tools, in the context of fractions
learning. Within the mathematics education research, the relationship between the mathematical
tools and the mathematical concepts has been studied under different theoretical framework, in particular Cultural Historical Activity Theory (CHAT)
and Actor-Network Theory (ANT). CHAT providing
a detailed theoretical lens to analyse the mediated
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Children’s perception of the affordances of the mathematical tools (Yasmine Abtahi)
actions of a subject, investigates the subjects activity in relation to a particular goal (Engestrom, 1999).
ANT, on the other hand examines the association of
human and non-human entities as nodes of a network
(Fenwick & Edwards, 2010) nodes of the network. Both
these theories, even though seemingly relevant, have
limitations in the particular context of my study. My
focus in this study is on how children, with their own
perceptions, utilise the physical properties of tools
to construct meaning for the mathematical concept,
through solving a mathematical task. Neither CHAT
nor ANT provides me with a lens to thoroughly examine the role of the physical properties of the tools in
children’s understanding of fractions. Therefore, to
look at the physical properties of mathematical tools –
as objects in their own right – I used Gibson’s notion of
affordances. And to look into children’s mathematical
perception as they interact with the tools (i.e., the interrelationship between the tools and mathematical
meanings). I introduce the concept of perception, as
viewed by Vygotsky in the object/meaning ratio.
GIBSON’S VIEW OF AFFORDANCES
Concerned with how the environment supports cognitive activity, Gibson (1977) contended, “in any interaction involving an agent with some other system,
conditions that enable that interaction include some
properties of the agent along with some properties
of the other system” (Gibson, 1977, p. 72). Gibson’s notion of affordance focuses on the contribution of the
physical system to the cognitive activity. The term
affordance refers to whatever it is about the environment that contributes to the kind of interaction that
occurs. In relation to the learning of mathematics,
the term affordances refer to whatever it is about a
mathematical tool that contributes to the interaction
of the child with these tools in the process of solving
a mathematical task. I consequently refer to the term
perception as whatever it is about the child’s thinking
that contributes to the interaction with the tools.
In a child’s interaction with the tools, it is crucial to
highlight where to locate the reference of the term affordance. For example, is the affordance that fraction
circles provide for making half a unit or 1/2, a property of the fraction circles, a property of the child interacting with it, or properties of both? Fraction circles
are a set of nine circles of different colours. Each circle
is broken into different equal fractional parts, which
use the same size as a whole. Gibson argued “affor-
dance is a property of whatever the person interacts
with, but to be in the category of properties, we call
affordances, it has to be a property that interacts with
a property of an agent in such a way that an activity
can be supported” (p. 341). Hence, for the properties
of a mathematical tool to be called its affordances, they
need to be perceived by the child in such a way that
a mathematical activity can be supported. For example, the physical properties of fraction circles that
might assist a child to grasp a fractional concept are
called their affordances only if the child perceives the
interrelationship between the physical components
of the fraction circles and the mathematical task, e.g.,
to solve 1/3 + 1/2. The physical properties of fraction
circles might also be perceived as useful to build a
bridge; these properties are called affordances if the
task at hand is bridge making.
The above argument implies that interacting with an
environment that provides an affordance for some
activity does not entail that the activity will happen;
the occurrence of the activity is intertwined with:
the activity of the agent in that situation – that is his/
her perception – and the task at hand. Assuming the
task at hand is a particular mathematical task to be
solved by the child, it is then the perception of the
child that needs in-depth analysis; that is how the child
perceives the tool and its affordances, in accordance
with the mathematical task. Vygotsky’s notion of
Object/Meaning ratio offers a systematic approach
to look at the gradual yet complex process of change
in the child’s perception as she/he interacts with the
mathematical tools to make meanings for the tool as
well as the mathematical concept.
VYGOTSKY’S VIEW OF PERCEPTION
A special feature of human perception is the perception of real objects (Vygotsky, 1976). The perception
of the real objects involves the perception of not only
colours and shapes but also of meaning; we do not
see a round object with two hands, we see a ‘clock’.
The attachment of meaning to an object is a process
that develops through the use of signs in interactions
with tools (Vygotsky, 1978). Signs, such as language,
drawings and the various systems of counting, are
‘means of internal activity aimed at mastering oneself’
(Vygotsky, 1978, p. 55). To better explain the process
of attaching signs to the use of tool-like objects, I use
Vygotsky’s object/meaning ratios.
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Children’s perception of the affordances of the mathematical tools (Yasmine Abtahi)
Vygotsky argued that at first the perception of a human being could be expressed figuratively as a ratio
in which the numerator is the object and the meaning
is denominator – object/meaning. This means that for
a young child the object is dominant and the meaning
of the object is subordinate. At this stage, the physical
properties of things play an important role in a child’s
interaction with them. For instance, a stick can be a
horse in child’s play but a box of matches cannot be
a horse. It is only later, when the child can make use
of signs and symbols in her/his interaction with the
objects, that the meaning becomes the central point
and objects are moved from being dominant to being
subordinate, thus giving rise to the meaning/object
ratio. At this stage Vygotsky noted that, for example,
to show a location of a horse on a map a child could
put a box of matches down and say, ‘This is a horse’.
The perception of the child can now be expressed as
a meaning/object. This figure of perception in which
the meaning dominates is the result of tying signs
to the tools; the box of matches is a symbol (sign) to
represent the horse.
Vygotsky’s object/meaning view has a two-fold theoretical implication for the study of children’s perceptions as they interact with the mathematical tools to
solve a task. On the one hand, it provides a base for
analysing the gradual changes of the child’s perception of the affordances of the tools as the child interact the tools to work on a mathematical activity. On
the other hand, it provides a base for analysing the
gradual changes of the child’s perception of the interrelationship between the tools and the mathematical
meaning they are intending to represent.
Provided that the child is interacting with a mathematical tool to solve a mathematical task, at the initial
stage of the child’s encounter with the mathematical
tool, the child’s perception can be presented by the
object/meaning. This figure of perception applies to
how the child perceives the affordances of the tool
(i.e., the meaning of the tools as an object in relation
to the task at hand) as well as how the child perceives
the mathematical concept represented by the tool (i.e.,
the mathematical meaning). At this stage, the tool is
dominant and its meaning(s) – as an object or its mathematical meaning – is subordinate. Hence, this is the
stage that the physical properties of the mathematical
tool play an important role in the child’s interaction
with them, both to perceive the affordances provided
by the tool and to perceive the mathematical concept
presented by the tool. For example, in fraction circles, the relationship between sizes and colours of
the pieces plays an important role in how the child
perceives the affordances of the fraction circles and
the fractional concepts they are presenting.
In order to invert this ratio, that is, in order for a tool
to be used as a symbol (a sign) for a mathematical concept, the child needs to increasingly tie signs to their
use of the tool. Children do this by talking about what
they do, talking about the tasks, drawing, and using
mathematical symbols. It is in the gradual process
of inverting the object/meaning ratio to a meaning/
object ratio that children grasp the interrelationship
between the affordances of mathematical tools and
the meaning of the mathematical concepts that they
are intending to represent.
In the following sections, I illustrate these ideas with
two examples of children’s interactions with mathematical tools as they attempted to solve addition
of fractions problems. In the first example, N and
J used their perception of the addition of fractions
to make meaning of the affordances of a newly designed mathematical tool (i.e. the fraction board) in
relation to the task. During the interviews, N and J
used different mathematical tools, such as fraction
strips and Cuisenaire Rods to solve different addition
of fractions problems. My rationale to report their
interaction with the fraction board is that this tool
was designed by me, hence children had no previous
encounters with the tools. Children’s first interaction
with the tools gave me an opportunity to examine how
they used the mathematical perception of the addition of fractions to perceive the affordances of the
fraction board. In the second example, Teresa used
the affordances provided by a mathematical tool to
make meaning of the addition of fractions. In both
cases, however, the children’s perception, both of the
mathematical concept and of the tool, changed gradually and through tying signs to their interactions with
the tools as they worked on the specific mathematical
tasks. The reason for selecting these two pieces of data
is to implicitly illustrate how the children’s perceptual
change of the affordance of the mathematical tools
goes through similar gradual and complex process
as children’s gradual changes in perception of the
mathematical meanings presented by the tool.
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Children’s perception of the affordances of the mathematical tools (Yasmine Abtahi)
THE CASE OF J AND N: WORKING
WITH THE FRACTION BOARD
J and N, in grade 5, participated in a small-scale research study in which they were asked to use a mathematical tool called the fraction board to solve 1/6 + 2/5.
The fraction board is designed to help students with
the addition of fractions. It contains fraction strips
of half, third, fourth, fifth and sixth, a board which
frames a fraction chart from one to 1/30, and a wooden
roller which holds the fraction strips and moved up
and down the board (Figure 1).
Both N and J has previously demonstrated an understanding of the addition of fractions through other
tasks, but neither of them had had previous encounters with the fraction board. Hence, they initiated the
task by attempting to grasp the mathematical affordances of the fraction board, as an object on its own
right. With the initial help of the researcher (Y), N and
J started to perceive the general physical properties
of the tool, not in any particular relation to the task
at hand. For example when Y began to introduce the
physical components of the board both N and J quickly
perceived the affordances of the strips by assigning a
fractional amount to them:
Y.
J.
Okay I just very quickly tell you that this
is a fraction board. These are called fraction strips and we have different kinds
This is 1/2 and this is 1/4 [pointing to different fraction strips]
At this stage, the physical properties of the tool played
an important role in N and J’s perception of the tool.
For example, the ways in which different fraction
strips are partitioned into different equal sized
parts, assisted N and J to perceive their affordances.
Through the gradual increase of the use of signs in
their interaction with the tool, N and J’s perception
of the affordances of the strips gradually changed.
This time in relation to the particular task at hand
(i.e., to solve 1/6 + 2/5), they picked the strips of 6ths
and 5ths as stated:
J.
N.
that is the 1/6 so I need one of that
yes and for that you need two (pointing
to the 1/5 fraction strips).
[J colours the strips, one part on the 6ths and 2 parts
on 5ths]
It was only later, that N and J were able to tie signs
related to the mathematical meaning of the addition of
fractions to their use of fraction board, perceive other
affordances of the tool, and solve the mathematical
task at hand. After selecting the two useful fraction
strips of (6ths and 5ths). N and J started to perceive
the “adding” affordances of the tool: “N. and I …I…I
guess they would go in here (pointing to the roller)”.
After loading the strips on the roller, N and J did not
immediately perceived the affordance that the tool
provided for finding the common denominator. So
they used their perception of the mathematical concept of the need of a common denominator to perceive
the affordances of the tool. They knew that to add 1/6 +
2/5, they needed to “turn the 5 and 6 into something”.
So they started to randomly moving the roller down
the fraction board to see what number on the chart fit
in both 1/6ths and 1/5ths. They unsuccessfully tried
12ths, 9ths and 18ths:
J.
we can turn this (6ths) into two twelve’s.
[they stopped at the 12ths line, noting 5ths
did not fit]
N.
Lets try the nines
J.nop…
N.
No definitely not
L.
try 18 … 18 going to 18?
After a few trial and errors they again used their
perception of the concept of addition of fraction to
conclude that the number that fits in both 5ths and
6ths is the thirtieths.
Figure 1
J.
[…]
J.
N.
J.
Oh… I get it, I can do it.
1, 2, 3, 4, 5
five… five thirtieth
so we can trade this into five thirtieths
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Children’s perception of the affordances of the mathematical tools (Yasmine Abtahi)
N.
J.
and then counting on the sixths… (counting from the top of the roller…), so 1, 2, 3,
4,5, 6,7 ,8, 9, 10, 11, 12… twelve thirtieth
so we can trade this (pointing to the one
sixth strip on the fraction board) with
12/30. So we can change it into 17/30.
N and J’s interaction with the fraction board is an
example of how perceiving the mathematical affordances of a mathematical tool is a gradual process,
which is intertwined with the mathematical task.
Moreover, this example shows how N and J’s object/
meaning ratio (where they interacted with physical
properties the fraction strips and the board to perceive the affordances of the fraction board) was inverted to a meaning/object ratio (where they used the
affordances of the fraction board to solve the task).
THE CASE OF TERESA: WORKING
WITH FRACTION KITS
This example is borrowed from Pirie and Kieren’s
(1989) study, in which Teresa used fraction kits to solve
an addition of fraction problem. The reason for including this episode is that Pirie and Kieren’s (1989)
study was conducted over a period of time, which
made it possible to look at Teresa’s gradual changes in perception over a longer time span. Fraction
Kits were designed by Tom Kieren; they contain rectangles, based on a common standard sheet as a unit,
representing halves, thirds, fourths, sixths, eighths,
twelfths, and twenty-fourths.
Teresa began the task of adding two fractions, not
knowing what to do. She said ‘I don’t know’ and ‘I think
you just add the tops and the bottoms’ (Pirie & Kieren,
1989, p. 163). She was then given the fractions kits and
a series of tasks. By perceiving the affordances provided by the fraction kit, ‘she noticed that one fourth,
three eights, and two sixteenths together exactly cover three fourths’ (Pirie & Kieren, 1989, p. 163). Later,
Teresa could ‘add’ 1/3 + 1/6 + 6/12 using the affordances
of the kit. This is the stage in which the physical properties of the fraction kit, for example, the relationship
between the sizes and colours of the pieces, played
an important role in Teresa’s interaction with the kit.
After a while, with a gradual change in her perception of the mathematical concept of the addition of
fractions, she was able to tie signs to her interaction
with the tool: ‘You can do 2/3 + 5/6 because twelfths
fit on both’ (Pirie & Kieren, 1989, p. 167). Later, when
asked ‘What is 1/2 + 3/4 + 2/5 + 7/10?’, Teresa, without
using the kit, she said:
Twentieths will fit on all of them. Two times ten
makes twenty, so one times ten or ten twentieths.
Four times five makes twenty so three times five
is fifteen twentieths... (p. 169).
Teresa’s gradual perceptual development for the addition of fractions, through the use of the fraction kit,
made it possible for her to make statements like:
Addition is easy. You can make up the right kind
of fractions just by multiplying the denominators
and then just get the right numerators by multiplying by the right amounts (p. 169).
This example shows how in the process of Teresa’s
interaction with the mathematical tool her object/
meaning perception (where she used the fraction kit
to solve the task) was inverted to become a meaning/
object ratio (where she used signs and symbols to solve
the task). In this process, Teresa increasingly used
signs, while interacting with the fraction kits.
DISCUSSION
In this paper, I have employed Gibson’s concept of affordances and Vygotsky’s notion of object/meaning
ratio to analyse children’s interactions with two different mathematical tools. Teresa’s case demonstrated
a gradual perceptual change in the mathematical concept of the addition of fractions as she used a fraction
kit to solve the addition of fractions tasks. J and N’s
case, by contrast, showed a gradual perceptual change
in the affordances of the mathematical tool (i.e., the
fraction board). These two examples show how children’s perception of a mathematical tool and their perception of the mathematical meanings presented by
the tool go through similar gradual and complex processes. Moreover, in both cases the children perceptions, of the tools and of the mathematical meanings,
are highly intertwined with the children’s attempt to
solve the mathematical task. For example, in J and N’s
case, they would have perceived different affordances
provided by the fraction board, had the task been to
find the equivalent fractions of a particular fraction.
Perceiving the affordances of a mathematical tool is
highly intertwined with perceiving the mathematical
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Children’s perception of the affordances of the mathematical tools (Yasmine Abtahi)
concepts there are intended to represent and with the
task at hand. However, the main reason for contrasting the case of Teresa (implicit focus on changes in
perception of mathematical meaning) with the case
of N and J (implicit focus on changes in perception on
the affordance of the tool) in this paper, is to illustrate
significance of combining the Gibson’s view of affordances and Vygotsky’s view of perceptual changes.
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Based on Gibson’s view, the affordance of an object
only become apparent in the ways in which the object is being used in a particular task. Based on this
perspective the meaning of the object is interrelated
with not only what the child is doing with the object,
in relation to the task at hand, but also the mathematical meanings that may or may not be apparent to the
child using the tool. Consequently, the Gibson’s view
of an object makes Vygotsky’s notion perception in
the object/meaning ratio more dynamic.
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ratio may shed some light on how the children use
the physical properties of the mathematical tools to
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