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Interactions between pupils’ actions and manipulative
characteristics when solving an arithmetical task
Doris Jeannotte, Claudia Corriveau
To cite this version:
Doris Jeannotte, Claudia Corriveau. Interactions between pupils’ actions and manipulative charac-
teristics when solving an arithmetical task. Eleventh Congress of the European Society for Research
in Mathematics Education, Utrecht University, Feb 2019, Utrecht, Netherlands. �hal-02400990�
Interactions between pupils’ actions and manipulative characteristics
when solving an arithmetical task
Doris Jeannotte1 and Claudia Corriveau2
1Université du Québec à Montréal, Canada; doris.jeannotte@uqam.ca
2Université Laval, Canada; claudia.corriveau@fse.ulaval.ca
In this paper, we explore the use of manipulatives in the classroom to solve an arithmetical task via
the concept of affordance. Manipulatives are part of the elementary class culture in different
countries, and even if some studies question the efficacy of manipulatives, there seems to have a
consensus around the necessity of using it. However, little is known about how mathematics is done
with manipulatives. The analysis of pupils’ actions helps put to light different affordances of two
manipulatives: base-ten blocks and abacus, in a classroom setting where the operations of addition
and subtraction are explored with pupils.
Keywords: Elementary School Mathematics, Arithmetic, Manipulatives, Affordance.
Introduction
Research and mathematics education communities at elementary level agree that using
manipulatives promote pupil learning (Moyer, 2001). Some researchers see the use of manipulatives
as a less abstract way of reasoning than with formal mathematical symbols (Lett, 2007, Özgün-
Koca, & Edwards, 2011). According to Domino (2010), Piaget (1964), Bruner (1977) and Diènes’
(1973) theories support most of the work on the uses of manipulatives. These theories of learning
are based on the idea that manipulatives are necessary for the development of mental images that
pupils may eventually summon in situations without manipulatives. In a way, the need, more over
the benefit, of manipulatives seems to be taken for granted by many researchers. It leads to many
other assumptions such as “doing math with manipulative is a concrete version of doing math” or
“manipulatives should support the construct of mathematical objects”. In our research about
manipulatives uses, we challenge these assertions. Doing math with or without manipulatives aren’t
the same activity. Based on this premise, we developed a project called MathéRéaliser, in which,
we want to understand what it is to do mathematics with manipulatives in a school setting.
Indeed, research shows that the context used to do mathematics structures mathematical activities
(e.g. Lave, 1988; Nunès, Schliemann, & Carraher, 1993). These studies, mainly conducted in
nonacademic contexts, highlight the situated aspect of mathematical knowledge (Noss, 2002). For
example, Pozzi, Noss, and Hoyles (1998) mention that in the professional context (nursing,
banking, engineering, etc.), the tools and objects available to actors shape their mathematical
activity. In other words, mathematical reasoning is developed in coordination with the “noise” of
the situations in which it takes place (Noss, 2002). In this perspective, doing mathematics with
manipulatives could be something very different from a concrete version of doing mathematics
without them (Corriveau & Jeannotte, 2015). Kosko and Wilkins (2010) were able to show that the
use of manipulatives colors the mathematical discourse developed with it. This aligns with a
sociocultural point of view where mathematical learning is shaped by the historical culture of the
community where the learning takes place and by the learners’ culture and experiences themselves
(Sfard, 2008). Manipulatives when available during a mathematical activity bear a certain culture
and shape the learner experiences.
In this paper, we aim to better understand what it is doing mathematics with manipulative in a
calculation task at the elementary level. To do so, the concept of affordance is convoked to identify
the potentialities of manipulatives in developing number sense through arithmetical operations.
Conceptual Clarification
Manipulatives and arithmetic
In the context of arithmetic, a plethora of manipulatives are available for elementary school. Poirier
(2001) classified manipulatives used for developing number sense in three categories. The first
category refers to manipulatives where units are visible and accessible. For example, if we use 3
free tokens and four transparent bags of 10 tokens to represent the number 43, the tens, represented
by bags that contain 10 tokens, can be “broken” into 10 units (by ungrouping a bag). The second
category refers to manipulatives where units are still visible but not accessible (not breakable).
Base-ten blocks are a good example of this kind of manipulatives: we have to physically exchange
one long for 10 units because the longs are usually unbreakable. The third category refers to
symbolic manipulatives, where units are not visible in tens, nor in hundreds, etc. (e.g. abacus,
money).
Furthermore, manipulatives can become a support or a constraint for the pupils’ mathematical
activity. On the one hand, it can support pupils reasoning. For example, base-ten blocks may help to
explain why an algorithm works. On the other hand, it can be considered as a constraint if we
impose a certain manipulative to solve a task (Jeannotte & Corriveau, 2015). By imposing
manipulatives, the task may gain in complexity.
Affordance
Affordances is linked to interactions between an individual and the environment. Environmental
characteristics (classroom setting, properties of the manipulatives, etc.) cannot be detached from
pupils (their perceptions, their experiences, etc.). They form an inseparable pair. According to Clot
and Béguin (2004), affordances are characterized, on the one hand, by the fact that objects are
significant, the user’s experience relates on this signification. On the other hand, by its praxis value:
“an object is immediately associated with a signification for action” (p. 53). Also, “[w]hether or not
the affordance is perceived or attended to will change as the need of the observer changes, but being
invariant, it is always there to be perceived” (Gibson 1977, in Brown et al., 2004, p. 120). So,
depending on the needs, the uses and the properties exploited of manipulatives may vary. e.g., the
base-ten blocks have been designed to help pupils perceived the structure of our numeration system.
As an adult, we can see these properties and exploit them to expose some mathematical patterns.
“We are seeing concepts that we already understand” (Ball, 1992, p. 5). What pupils see is also
related to what they know. Since manipulatives aren’t solely used by pupils, but also by teachers,
what teachers do with manipulatives is also inherently part of pupils’ experiences. Thereby, the
concept of affordance can not only offer an insight on learning processes but also on teaching
processes.
Methodology
For the aims of this paper, we focus only on a part of the data collected from the MathéRéaliser
Project. We conducted a collaborative research that solicits the participation of teachers. As
Corriveau and Bednarz (2016) uphold:
[t]his perspective leads us to re-think […] the relations between the researcher, as being the
expert, and the teacher, as being the novice or the user, as frequently conceived in research in
mathematics education […] [T]he teacher is assumed to be reflective and knowledgeable (pp. 1–
2).
In collaborative research, the concept of “double relevance” (Desgagné, 1998) is fundamental and
refers to the construct of argumentation relevant to both communities, research and practice. A
“double relevance” underlines every choice made in the collaborative work. For example, if a task
is chosen to be experimented with pupils, it means that it makes sense for teachers and researches
according to their respective sensible arguments.
Description of the Task
The task we experimented is inspired by Cobb (1994). Three similar questions to the following
were answered by the pupils. The task was presented orally.
“Represent 1009 with your manipulative and then, I’ll ask you a question. [Wait for the
pupil to represent 1009 with base-ten blocks or homemade abacus.] I had a number, I
subtracted 453. [Write it on the blackboard.] I now have 1009. How much did I have at the
beginning?”
Description of the co-constructed Lesson
The task was experimented with two grade 3 classes (9–10 years old) of 19 and 18 pupils. They
already had learned an algorithm using drawing of base-ten blocks. They were not familiar with the
abacus. Pupils worked by two using either base-ten blocks or homemade abacus. After teamwork, a
whole class discussion took place about 1) the different answers obtained and 2) the strategies used
to resolve the task. Table 1 presents the double-relevance of this task and its experimentation.
Plausible use of the task for teachers
Relevance of the task for researchers
Task congruent with curriculum expectations (choice
of operation, adding and subtracting 3–4 digit
numbers).
Curriculum fosters the use of manipulatives.
Opportunity to reflect on the use of manipulative by
their pupils.
Manipulatives seen as a support to solve the task.
Task that could lead to mathematical argumentation
(see Cobb, 1994).
Manipulatives seen as a constraint in this particular
task. Since pupils used to operate by drawing base-
ten blocks, asking them to operate with real base-ten
block or abacus is unfamiliar to them. This
introduction of foreign elements may serve as a
breaching experiment (Garfinkel, 1967) to put to
light usual ways of doing.
Table 1. Double-relevance of the task
Characteristics of the manipulatives
To perform the task, two manipulatives were available. Either pupils worked with base-ten blocks
or with a homemade abacus (see Figure 1).
Figure 1: Base-ten blocks and homemade abacus
As each manipulative has his own properties, we can conjecture that each manipulative, in relation
with pupils and the whole class generates different affordances in the classroom (see Table 2).
Base-ten blocks
Homemade abacus
Proportional model;
Units visible but not accessible (not
breakable);
The value does not depend on the
arrangement of the manipulatives;
The same manipulative “object” can only
take one value.
Non-proportional model;
Symbolic manipulatives (units not visible
nor breakable);
The value depends on the arrangement of
the manipulatives;
The same manipulative “object” can
change value.
Table 2: Characteristics of each manipulatives
Analysis and Results
Data analysis has involved three stages. First, to explore the affordance in the use of manipulatives
by the pupils, we watched the videos multiple times (Powell, Francisco, & Maher, 2003). Secondly,
we extracted every pupil’s actions we observed. Every action was then described in association with
the characteristics of the manipulatives. Finally, we grouped actions related to three different
mathematical activities involved with the task and the classroom setting (the use of manipulatives).
One may think that the main difficulties of this task is to choose the good operation and we are
aware that using manipulatives do not help the pupils with this choice. However, in this
experimentation, it did not appear as an issue. Moreover, the use of base-ten blocks and the abacus
helped the teacher to see at a glance who chose the right operation. Actually, most errors arose from
counting strategies. Table 3 presents the analysis of pupils’ actions in relation to the characteristics
of each manipulatives. We grouped the different actions according to the mathematical activity
involved: representing, operating and interpreting.
When representing numbers, most pupils were able to exploit the manipulatives. For base-ten
blocks, they associated the right blocks (units, longs, flats, etc.) to each position. Nevertheless, few
of them disposed the blocks in a way they can “recognize” rapidly the number in front of them. For
the abacus, some pupils struggle with choosing the right column when representing the second term
of the operation. Furthermore, more pupils relied on visual patterning, thus helping them later
interpreting the number.
When operating, other than counting mistakes, difficulties arose from converting strategies. Even if
pupils referred to changing 10 ones for 1 ten (for example) in their discourse, we observed more
than once pupils changing eleven units for 1 long. When using the abacus, to convert a ten into
ones, they move one chip from the tens column to the ones column AND add ten new chips in the
ones column.
One pupil worked with the abacus inconsistently: he represented one thousand nine from right to
left, he managed the calculations, but when interpreting the result, he read the number from left to
right. This gave the opportunity to the teacher to talk about communication in mathematics.
Maths
Activities
Base-ten blocks characteristics and actions
observed
Homemade abacus characteristics and
actions observed
Representing
Counting the blocks needed for each
“place value”: cubes, flats, longs, units.
Disposing the blocks so the highest
value is far left and the other ones on its
right (no empty spaces for zero values).
Piling the blocks (e.g. 9 units on a cube
represent 1009).
Using the space to differentiate
positions: e.g. ten flats (instead of a
cube), then the other flats, etc.
Counting correctly or not the chips and
disposing them in the right or wrong
column.
Disposing the chips in the same order we
read numbers. Only one pupil disposed it
in the opposite order.
In each column, using or not visual pattern
disposition when placing chips in each
column (e.g. 3 rows of 3 to represent 9).
Operating
Algorithm
Place value dealing
o Dealing with positions where no
exchange is required first
o Operating from left to right
o Calking the taught algorithm (from
units to cubes)
o Mixed strategies
Intermediary calculations
o Counting (e.g.: when adding 4 longs
to 8 longs, regroup all longs and
count them)
Place value dealing
o First, dealing with columns where no
exchange is required
o Using the taught algorithm (from ones
to thousands)
Intermediary calculations
o Counting (e.g.: when adding 4 chips to
8 chips, regrouping all the chips in one
column and count the result)
o Mental math, articulating
manipulatives and number facts
correctly or not.
Converting
Changing one long into ten units (or ten
units into one long)
o by counting correctly or not ten
units;
o (from ten units to one long) by
removing all the units counted or not;
o changing eleven units for a long.
Changing one chip into ten chips in
another column (vice versa)
o by counting correctly or not ten chips
o (from one ten to ten ones) by moving
one chip from the tens to the ones
column and adding 10 new chips.
o (From ten ones to one ten) by
removing all the chips counted or not.
By placing in the right or wrong column
the chips exchanged.
Interpreting
Associating the right value to each sort
of block.
Associate the right column to each
position
Changing the reading direction from the
one used when representing
Table 3: Actions made by pupils in relation to the characteristics of each manipulatives
Discussion
With the table presented above, we have tried to better understand what it means to do mathematics
with manipulatives when developing calculation abilities at the elementary level. By examining the
characteristics associated with action, we put to light different affordances. Even if some
affordances are shared, some others are specific to only one of the manipulatives. In this section, we
discuss further one idea that emerged from our analysis. This idea allows to put to light some
differences in pupils’ actions that are related to manipulatives characteristics. Also, speaking of
affordance is speaking of seeing what could be observed but is not. Both, what is observed and what
is not can inform of the manipulative practices in the classroom.
Counting Over and Over
In the first grades of the elementary, children are used to count to solve problems. The teachers then
try to enrich their number sense and to complexify their counting strategies (with number facts,
adding to ten, visual pattern disposition, etc.). However, for most pupils, the counting aspect
brought by the use of manipulatives clearly took over other calculation abilities and some control
was lost in their mathematical activities. When using base-ten blocks, most pupils counted the
blocks even for small quantities. For example, when adding three to nine units, we observed pupils
counting three units, then counting nine units, putting them together, counting the total, obtaining
twelve units, counting ten units to trade them for one long. This may seem trivial insofar as the
counting strategy is achieved with success; it does not mean there is a lack of control. Nevertheless,
we have observed several errors that arise from this way of doing even if pupils know their
numerical facts. For example, a pupil took eight units, but counted nine. Thus, when he added three
units and counted the total, he obtained eleven units (and not twelve). He continued his calculations
with eleven units and obtained a very close answer, but the wrong one. However, we observed him
mentioned to another pupil, further in the video, that nine and three give twelve.
Also, even if pupils are able to group by tens and ungroup, they do not rely on this property yet
visible in the base-ten blocks. They rather count. For example, again, to add three to nine units,
pupils could have taken one long and kept only two units directly, but they count as we described it
above. However, while we observed the same way of doing with an abacus, we also saw more
complex strategies. For example, when adding nine and three, a pupil added to ten. However, he
was not able to coordinate his action with the abacus and the mental operation.
In short, the main action observed during the task was counting. Counting over and over seemed to
divide the global task into counting sub-tasks. After each sub-task, most pupils had to think again
about what they were doing sinking into a vicious cycle: thinking about the global task made them
forget about the counting they did and counting again made them forget about the global task. With
manipulatives, it is difficult, for pupils, to keep track of what is done and what has to be done.
Does this mean that using manipulatives only suggest this way of doing things for those pupils?
There is more than one answer to that question. On the one hand, it does because even if it is not
necessary all the time, there is always some counting to do. Using manipulatives to add or subtract
involve counting physical objects. The physical aspect of manipulatives asks for this way of doing.
However, counting is not always the most efficient strategy to use here. Of course, the environment,
i.e. the usual class activities, certainly contributes to reinforcing this strategy. On the other hand, as
mentioned years ago by researchers (e.g. Bednarz & Janvier, 1982), the tasks performed with the
help of manipulatives relate essentially to number representation and “translation”, a work on the
number representation in our numeration system. As pupils presumably used base-ten blocks to
count in the first place and less to operate coordinating other strategies as well, we can think that the
praxis of this manipulative associate it immediately to counting. While we tend to believe that the
use of manipulatives supports mathematical reasoning, in this case, we rather observed that pupils
refer to more basic reasoning than they could have used.
In fact, we observed more advanced strategies with the less familiar abacus than with base-ten
blocks used since first grade. For McNeil & Jarvin (2007), working with familiar manipulatives
might drive the attention of pupils in the wrong. In our case, it is not a wrong direction, but it seems
to lead to more basic strategies and to some lack of control. We could conjecture that using a less
familiar manipulative does not drive pupils to usual uses. Indeed, in terms of affordance, the
meaning constructed in the interactions between the abacus and pupil is less constraint by the
experiences. It gives leeway to integrate skills in a new situation (e.g. visual patterning, use of
number facts, etc.).
Conclusion
Carbonneau, Marley & Selig (2013) said that “specific instructional variables either suppress or
increase the efficacy of manipulatives suggests that simply incorporating manipulatives into
mathematics instruction may not be enough to increase student achievement in mathematics”
(p. 397). Looking at affordance help understand that not only the instructional setting, but the
culture of the class and pupils’ experiences also play a role when talking about learning with
manipulatives. As manipulatives used to solve this task are by definition physical object, counting
can’t be detached from it. Doing math with manipulatives was not the same activity in this
particular case. We observed that pupils rely mostly on what they used to do with a specific kind of
manipulatives rather than what they used to do without them. The question is then how to help the
pupils rely on other characteristics of these manipulatives? Drijvers (2003) stressed that the
affordance that could be realized in the classroom rely not only on the tool itself, but rather on the
exploitation of these affordances the educational context and the teacher drive that. As base-ten
blocks are often used in grades 1 and 2 when pupils don’t master the table facts, the teachers’ role is
here quite important. They have to help pupils go beyond counting and perceived the power of
organizing manipulatives by exploiting the space to “see” table fact and not solely rely on it to
count and to represent numbers. To do so, teachers could organize the educational context around
manipulatives to provide wider learning opportunities for the pupils.
Acknowledgment
The Canadian Social Sciences and Humanities Research Council fund MathéRealiser.
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