ZDM Mathematics Education (2014) 46:95–107
DOI 10.1007/s11858-013-0550-2
ORIGINAL ARTICLE
How visual representations participate in algebra
classes’ mathematical activity
Maria Manuela David • Vanessa Sena Tomaz
Maria Cristina Costa Ferreira
•
Accepted: 1 October 2013 / Published online: 17 October 2013
FIZ Karlsruhe 2013
Abstract Our aim is to discuss how a visual display
introduced in a classroom activity to represent a specific
algebraic procedure is transformed, taking a central role
and modifying the ongoing activity. To discuss how visualization comes about in this activity, we describe an
illustrative example selected from observations carried out
in a 9th grade classroom and analyze the class interaction
from a cultural-historical perspective. Our analysis illuminates the tensions that emerge from a difference between
the teacher’s way of signifying the algebraic procedure and
the students’ overuse of a visual display they associate with
it, and how these tensions impel changes in the activity. We
further discuss some pros and cons of using visual displays
in algebra classes, and we argue that it is very important for
the teacher to be aware of them in order to realize the
benefits of using such displays.
Keywords Visualization in mathematics education Visual display Teaching of algebra Activity theory
1 Introduction
There is a long tradition of research on the role of visualization in mathematics education, encompassing a variety of
perspectives (Krutetskii 1976; Parzysz 1988; Zimmermann
M. M. David (&) V. S. Tomaz M. C. C. Ferreira
Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
e-mail: [email protected]
V. S. Tomaz
e-mail: [email protected]
M. C. C. Ferreira
e-mail: [email protected]
and Cunningham 1991; Bishop 1991; Fischbein 1993;
Casselman 2000; Arcavi 2003; Presmeg 2006; Duval 2006;
Bartolini Bussi and Mariotti 2008; Rivera 2011). The number of contributions in this area is so great and the variety of
perspectives is so vast, that a comprehensive review of the
literature is beyond the scope of this paper.
One such contribution that is relevant to our discussion
presents the idea of embodied mathematics. As Núñez
(2006) and Freitas and Sinclair (2012) point out, only
recently have researchers begun to investigate the role of
visually perceptible gestures to construct and communicate
mathematical meanings. However, in spite of the fact that
this is a relatively recent area of research, the complex
relations between gestures and speech, symbols, diagrams,
and other visual displays have already inspired some very
interesting and profound discussions (Núñez 2006; Radford
2003; Sinclair and Tabaghi 2010; Freitas and Sinclair
2012) that are of great relevance to our study. We rely on
these studies to explain that words, mathematical symbols,
gestures, diagrams, and other visual displays are more than
external representations of abstract mathematical concepts
or procedures, because they are inseparable parts of what
they are supposed to represent.
Although some difficulties, conflicts, tensions, and
obstacles related to the process of visualization are
mentioned in the literature in this area (Parzysz 1988;
Fischbein 1993; Mesquita 1998; Arcavi 2003; Duval
2006; Rivera 2011), the majority of the studies reviewed
stress how speech, gestures, drawings, diagrams, and
other visual displays all have an important role in the
visualization of mathematical concepts and procedures
and, therefore, also in the process of teaching and learning mathematics.
We share a common aim with this previous work in that
we also want to stress the very important role of
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visualization in mathematics education. However, we take
a different perspective, which we have already explored in
other work (Tomaz and David 2011; David and Tomaz
2012), by focusing on some tensions that tend to arise in
the process of visualization as it takes place in a classroom
geometrical activity and further discussing how these tensions impel changes in the activity as well as how they can
be attenuated.
As illustrated by David and Tomaz, the visual representations of the notions and/or relations between concepts
and ideas may trigger tensions and contradictions between
those representations and the thing they are supposed to
represent. Radford (2006), reporting on Kant, presents a
distinction between geometric and algebraic objects, as the
geometric ones can be represented directly with drawings,
and the algebraic ones only indirectly through drawings,
diagrams, and other visual displays based on signs.
In this paper, we address visualization in algebra. To do
this, we find support in Arcavi (2003) to start with a very
broad notion of what visualization means in mathematics,
‘‘as both the product and the process of creation, interpretation and reflection upon pictures and images’’ (p.
215). Further, regarding specifically the domain of algebra,
we share with Rivera (2011) the view that visually drawn
constructions of some mathematical objects, concepts or
processes can effectively assist in developing what Mason
et al. (2009) name as a structural awareness of the corresponding abstract knowledge, despite being in an incomplete form.
We substantiate our discussion on the use of a visual
display1 with such a strong appeal that it turns out to be
overused by the students, and we analyze how it participates and shows its influence in modifying and changing2
the classroom activity. The situations considered are related to the resolution of equations involving the distributive
law of multiplication with respect to addition and subtraction at the middle school level, and the association of
this law with a visual display, named in Brazil as the little
shower,3 and from now on just the shower. At the beginning of our study, we considered the shower simply as a set
of curved arrows (two or more) used to express the steps to
be followed in order to apply the distributive law to a given
mathematical expression. For us, at this point, the shower
was associated with the following four ways of
1
We use the term visual display as a general term to refer to all sorts
of drawings and diagrams as well as to all graphic marks commonly
used to represent and give meaning to mathematical ideas. In these,
we do not include alphanumeric symbols and other frequently used
mathematical symbols.
2
Latour (2005) argues that objects (or things) too have agency and
can be considered as participants in the course of social action.
3
This is a widely, informally used expression in Brazilian schools:
chuveirinho.
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presentation: (1) the graphic marks
, indicating
a movement or some actions to be performed with a given
expression; (2) a hand gesture reproducing this movement;
(3) the spoken word shower, associated in the real world
with an artifact that distributes water; and (4) the mathematical distributive law aðb cÞ ¼ ab ac.
Initially, we considered the shower as a mediating artifact, that is, as a visual representation of a sequence of
coordinated actions to facilitate the application of the
algebraic law. However, as the analysis of classroom
events developed, we quickly understood that there was
much more to say about it, because there were many more
meanings encapsulated in and associated with those graphic marks than we first thought.
Thus, our main objective in the present paper is as follows: To discuss how a visual display introduced in a
classroom activity to represent a specific algebraic procedure comes to incorporate multiple facets and takes on a
central role in modifying the ongoing activity.
2 Theoretical framework
Our work is grounded in activity theory (Engeström 1987),
which is a sociocultural perspective of analysis that elects
the activity system as the unit of analysis.
According to Leont’ev (1978), an activity consists of a
group of people (subjects) engaged in the same goal, with a
direction for their work (object or motive of the activity).
The activity emerges from a necessity, which directs the
motives towards a related object. To satisfy motives,
actions are necessary. These actions, in turn, are performed
according to the conditions of the activity, which determine
the operations related with each action. Therefore, in the
structure proposed by Leont’ev, the activity, directed to a
motive, is located on the first level. On the second level,
there are actions directed to specific objectives, and on the
third level come the operations, or routines, that keep the
system functioning and are dependent upon the conditions
of the activity.
Engeström (1987) resumes and reformulates the structure proposed by Leont’ev, which, in turn, is based in
Vygotsky’s structure, to represent a collective activity
system, by adding new components to the previous structure. In Engeström’s structure, the subject consists of an
individual or group of individuals engaged in a unique
goal, whose agency is the focus of analysis; the object is
the ‘‘space problem’’ in which direction the activity is
developed; artifacts are mediating instruments, tools, and
signs; community refers to the people who share the same
object; division of labor is related to the horizontal division
of tasks and to the vertical division of power and status of
How visual representations participate in algebra classes’ mathematical activity
the community members; and the rules refer to the implicit
and explicit norms and conventions that regulate the
actions and interactions within the activity system.
Engeström (2001) argues that ‘‘object-oriented actions
are always, explicitly or implicitly, characterized by
ambiguity, surprise, interpretation, sense making, and
potential for change’’ (p. 134). He also stresses the central
role of contradictions as sources of change and development of human activity, and explains that contradictions
are more than problems or conflicts; they are ‘‘historically
accumulating structural tensions within and between
activity systems’’ (p. 137). Engeström and Sannino (2010)
explain further that:
Conflicts, dilemmas, disturbances and local innovations may be analyzed as manifestations of the contradictions. There is a substantial difference between
conflict experiences and developmentally significant
contradictions. The first are situated at the level of
short-time action, the second are situated at the level
of activity and inter-activity, and have a much longer
life cycle. They are located at two different levels of
analysis. (p. 7)
According to this explanation, we understand that contradictions have a much wider sense than the one usually
attributed to the notion of cognitive conflict, for example,
since contradictions go far beyond the relation between
subject–object of knowledge. However, cognitive conflicts
can develop as contradictions, if and when they are historically accumulated.
Contradictions can be internal to the activity systems,
but they may also be external, for example, ‘‘when values,
beliefs, or activities of one activity system conflict with
those of another’’ (Jonassen 2000, p. 108). Within a
classroom, contradictions may be generated by the students
or the teacher, or by a particular artifact or rule, or by any
other component of the structure of the activity. However,
since in this structure all components are in a close relationship with each other, contradictions should always be
seen as expansible to the other components and to the
activity as a whole.
Contradictions generate questioning of the practices by
the subjects, causing ruptures, which can originate expansive transformations of the activity, when tensions and
contradictions are overcome:
An expansive transformation is accomplished when
the object and motive of the activity are reconceptualized to embrace a radically wider horizon of
possibilities than in the previous mode of the activity.
(Engeström 2001, p. 137)
These ideas were originally developed to deal with
large-scale human activities, in which expansive
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transformations usually occur only after long periods of
development. This theoretical perspective is not very
widely used in classroom research. However, recently,
Engeström and Sannino (2010) have proposed the possibility of using them as an analytical tool to describe and
discuss short-term processes of classroom activity. In order
not to lose sight of the historicity of the activity in a more
general perspective, one should alternate the lens of analysis, at times focusing on short-term processes (zooming
in) and at other times distancing the view (zooming out)
from those short-term processes in order to see them as part
of a longer-term classroom activity. They explain these
large-scale and small-scale expansive transformations of an
activity using the idea of cycles of learning actions:
The logic of the expansive cycle is such that a new
cycle is assumed to begin when an existing, relatively
stable pattern of activity begins to be questioned.
Correspondingly, the cycle ends when a new pattern
of activity has become consolidated and relatively
stable. (…) Large-scale cycles involve numerous
smaller cycles of learning actions. Such a smaller
cycle may take place within a few days or even hours
of intensive collaborative analysis and problem
solving. Careful investigation may reveal a rich texture of learning actions within such temporally short
efforts. (Engeström and Sannino 2010, p. 11)
Since these authors are speaking from the standpoint of
research, examining studies based on the theory of
expansive learning, it is assumed that it is up to the
researcher/observer to determine if and when a relatively
stable pattern of activity begins to be questioned and a new
one starts to be consolidated. The criteria for defining the
starting and end points of a cycle are subordinated to the
analysis to be performed and to the judgment of the
researcher/observer. In the present work, we consider that
momentary modifications in the components of a relatively
stable activity may occur, as far as these modifications are
not disruptive enough to completely modify the overall
structure of the ongoing activity.
This notion of expansive cycles has already proved to be
quite appropriate to discuss what is going on in some shortterm classroom activities (Tomaz and David 2011; David
and Tomaz 2012). Furthermore, this perspective offers a
framework of analysis capable of the following: (a) capturing the complexity of the interconnected activity systems composing the overall classroom activity; (b) making
use of different lenses (zooming in, zooming out, outwards,
and inwards) to focus on specific activity systems or in
overall systems, for example, focusing on the activity of
the students and the teacher, or on a group of students, or
on the teacher activity; (c) capturing the historicity of the
classroom activity and the cultural embodiment of the
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activity systems, identifying moments of possibly small
changes and transformations in the components of the
ongoing activity, and illuminating tensions and contradictions between different cultural perspectives present in the
classroom.
In this paper, we direct our analysis to one activity
system formed by a constellation of interconnected classroom activities, all of them involving the shower in the
product of algebraic expressions. We focus on four shortterm classroom activities and analyze them historically,
illuminating the complexity of the ongoing process of
visualization of the distributive property, discussing this
process under the cultural historical perspective of activity
theory. Not restricting ourselves to a strictly cognitive
perspective of visualization, we aim at revealing how
visualization participates in and modifies the school
mathematical activity in a more comprehensive manner.
We go beyond the relation of the subjects with the mathematical concepts, procedures and ideas, intermediated by
visual artifacts, to consider several other relations between
different components within this activity system.
3 Method
To describe and analyze this activity system, as mentioned
before, we discuss some classroom events, selected from
data collected by one of the authors, which we thought to
be appropriate for the discussion on how visualization
occurs in a classroom algebraic activity. These data were
produced through participant observation of the classroom
activity registered in written notes and in video recordings,
and interviews with the teacher and some students.
In accordance with our theoretical perspective, when
capturing the classroom interactions, we were imbued with
the idea of socially shared cognition, as a collective construction of knowledge and meanings, developing from the
interactions between the individuals, or with other components of the activity, pervaded by the social context in
which they occur.
Brown and Cole (2000) explain in more detail how the
idea of socially shared cognition can be seen in agreement
with the cultural-historical perspective we adopt here.
Based on the assumption that cognition is distributed
among the participants, the artifacts, and the social institutions to which they belong, they conclude that, ‘‘to say
that cognition is socially shared is to say that it is distributed (among artifacts as well as people), and that it is
situated in time and space. Because it is distributed, and its
assembly requires the active engagement of those involved,
it is to some extent constructed’’ (p. 198). With this idea in
mind, we are more interested in pursuing the interactions
between the teacher and the class as a whole, and not so
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much in focusing on the activity of the individual students,
although not completely disregarding its importance for the
overall class activity.
The observations took place in a 9th grade classroom of
a public school in Brazil, from April to August 2012. For
this grade level, there were three 90-min mathematics
classes per week, and 28 classes were videotaped during
the research period.
The school is a prestigious public school, and the
admissions selection process is a lottery, resulting in a
student body of mixed social economic backgrounds and
heterogeneous mathematical experience. There were 25
students in the classroom, 12 girls and 13 boys, ranging in
age from 14 to 16 years old.
This was the teacher’s first year teaching regular
mathematics classes, just after finishing his Master’s
degree in Mathematics Education. However, he already
had almost 3 years of experience working with public
school students, aged 7–15 years old, on mathematics
projects. In an interview, he said that, as a middle school
student, he had learned algebra as a series of routine procedures to solve exercises and had been capable of doing
this without any problems. However, at the university,
when he had to handle many more procedures, he found it
difficult to decide if they still worked or not, because he did
not attach any meaning to them. Now, as a teacher, he
believes it is important to attach meaning to the procedures,
especially in this class, as he perceived that the students
were using them without understanding what they were
doing. For example, the students already knew many
strategies for solving first degree equations and employed
many informal expressions such as ‘‘when it moves to the
other side, it changes its sign,’’ ‘‘cutting down,’’ or ‘‘cross
multiplying,’’ which made the teacher feel uncomfortable.
He insisted on asking the students to explain why a given
procedure could be applied to a specific equation. The
students found it completely unnecessary to justify the
procedures based on properties of the operations, although
they made frequent mistakes when using those shortcuts.
The excerpts of classroom dialogues and the pictures of
the teacher’s writing and gestures that underlie our analysis
are presented in such a way as to enable us to comprehend
how the shower procedure takes a central role, modifying
the ongoing classroom activity.4
The subject being taught was algebra, particularly first
and second degree equations and fractional equations. For
our analysis, we did not find it necessary to consider all
classes on the resolution of equations, since our interest in
this paper is the use of the shower in the product of algebraic expressions. This does not mean that we do not
4
The teacher is identified in the photos, with his permission, but all
names mentioned have been changed.
How visual representations participate in algebra classes’ mathematical activity
recognize the influence of other classes’ algebraic activities
on the situations we are interested in, nor that we do not
consider their relevance for understanding the historicity of
those specific situations.
The first class episode analyzed took place on April 23,
2012, and the subject was the translation from word
problems to first and second degree equations, and how to
decide if a given number is a root of an equation. The
second class episode took place on April 24, and the subject was the resolution of equations by isolating the variable. The third episode occurred on April 26, when the
teacher decided to discuss the distributive law because he
had noticed, in the previous class, that many students were
misusing this property, applying it also for the multiplication of three factors. After that, the students went back to
equations’ resolution by isolating the variable. The fourth
class episode selected for our analysis took place on June
19. The students were asked to solve word problems and
exercises, involving fractional equations, individually.
Afterwards, the teacher called a student to present his
solutions orally, while the teacher wrote it on the
blackboard.
In the period of time between the third and fourth class
episodes, the teacher presented other methods for solving
second degree equations, algebraically and geometrically.
The resolution of these equations did not require the use of
the distributive law. In the sequence, on June 11, fractional
equations were presented to the class. Their introduction
raised once again the need for the use of the distributivity
and triggered a discussion about the meaning of simplifying expressions (a procedure called by students ‘‘cutting
down’’) since some students were not sure if they could, for
instance, simplify the x in expressions such as xþ3
x2.
For our purpose, we have elected as our unit of analysis
a system of four interconnected activities, each one related
to one of the four class episodes briefly mentioned before.
In spite of our unit of analysis being composed of shortterm activities, it is possible not to lose sight of the historicity of the activity system, because the four activities
are distributed over a longer period of time (about
2 months), and the classroom observations covered an even
longer period (5 months).
4 The use of the shower in the product of algebraic
expressions
In the present analysis, we adopt the perspective of activity
theory not only for structuring the observed practices, but
also for exploring its full potential as a powerful analytical
tool to further discuss the role of the shower and the
modifications introduced by it in the activity system. We
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focus our analysis on one activity system of interconnected
and interdependent activities, the use of the shower in the
product of algebraic expressions, formed by a constellation
of four activities: (1) emergence of the term little shower;
(2) association of the parentheses with the shower; (3)
overgeneralization of the use of the shower in the product
of algebraic expressions; and (4) connecting the shower
with the distributive law.
Although our unit of analysis is formed by a constellation of four activities, there is a relatively stable pattern
encompassing these activities that makes it possible for us
to consider the overall system, the use of the shower in the
product of algebraic expressions, as structured by the following components: Object: use of the shower in the
product of algebraic expressions; Subject: teacher and
students; Artifacts: among many others, the shower, word
problems, algebraic expressions and parentheses; Community: other teachers and students from other classes and
previous years, curriculum designers, parents and textbook
writers; Rules: among many other rules, the students should
understand and be able to justify all procedures performed,
they should not solve equations mechanically and they
should solve first what is inside the parentheses; Division of
labor: the teacher is the authority.
However, when we zoom in and inwards toward the
constellation of the four activities, we notice that, in spite
of the overall stability of our unit of analysis, there is a
great mobility within this stability. This can be perceived
by giving a description and characterization of the four
activities.
For the sake of clarity and economy, in this paper, we do
not give a full description of the complete structure of these
four activities, as our main aim is to discuss the use and
role of the shower in the activity system and its influence
on the teacher’s and students’ actions, in order to further
discuss the role of visualization in an algebra class.
4.1 Activity 1: Emergence of the term little shower
The first selected episode begins when the teacher introduces a word problem for which the students should find a
second degree equation associated to it: Which number
gives the same result when we divide 4.5 by it and when it
is subtracted from 4.5? Is there more than one such number? The students should use appropriate letters in order to
form algebraic expressions for the equation. For this
problem, the students arrive at the following expression:
4:5 ¼ x ð4:5 xÞ. The teacher asks them what they should
do in order to solve the equation and, at the same time, he
makes some gestures with his fingers, as if drawing arcs
over the expression (Fig. 1). At the same time, one student
says little shower. Immediately afterwards, the teacher
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M. M. David et al.
Fig. 3 Gesture indicating that 5 ðx 2Þis a single thing
Fig. 1 Gesture indicating the arrows
equation appears: 5ðx2Þ
¼ ðx2Þðx3Þ
x3 . The teacher asks
2
3
the class what to do first in order to solve it, and one
student says that they should ‘‘remove the parentheses.’’
The teacher asks for alternative ways to start solving it, and
in response, one student suggests finding the Least Common Multiple (LCM) of 2 and 3, which is the procedure
expected by the teacher. He accepts this suggestion, writes
on the board 6 ¼ 6 6, saying ‘‘six by two… three… three
2
times five… fifteen’’ and then fills it in with 15ðx2Þ
¼ 6 6.
6
Immediately, a student asks a question, beginning the
exchange shown below:
1.
2.
Fig. 2 Drawing curved arrows
draws curved arrows on the board (Fig. 2), making an
association with the student’s suggestion.
In this first episode, we consider an activity system by
itself, but we do not give a full description of its structure.
For the purpose of this paper, it is sufficient to say that the
object of this activity is the translation of a word problem
by an algebraic equation, the subjects are the teacher and
the students and, among the activity artifacts, it is possible
to highlight the shower, which was already known by the
students. At this moment, the students seem to use the
shower without being conscious of its relation with the
distributive law, and the teacher does not seem to be aware
of the lack of this association.
At this point, we do not perceive any tension evolving in
the classroom activity originating from the student’s suggestion. The teacher naturally accepts the student’s use of
the word shower.
4.2 Activity 2: Association of the parentheses
with the shower
In the second episode, the students are not having difficulties solving the equations, since the equations are all of
the first degree and do not require the application of the
shower, up to the point when the following algebraic
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3.
4.
5.
6.
7.
8.
9.
10.
Student: Don’t we have to multiply inside (the
parentheses)?
Teacher: Guys… this here [pointing to 5 ðx 2Þ] is a
multiplication… isn’t it? … Isn’t it a single thing?
(Fig. 3)
Kleber: Couldn’t we multiply inside, too?
Teacher: Listen! If we take out the parentheses, we
are going to have one fraction and the other
fraction… So we have a subtraction… Isn’t it?…
Then, we will have to do a multiplication first and
afterwards the second multiplication… won’t we?
Students: Yes…
Teacher: But here [pointing to 5 ðx 2Þ], we have a
multiplication… So we can do 6 divided by 2…
resulting in 3 and then… 3 multiplied by 5 is 15
and… Only afterwards we are going to distribute…
Kleber: Prof… Prof… look… the parentheses… 6 by
2… 3… 3 times x… 3x and 3 times 5…
Teacher: Look… here [pointing to 5ðx 2Þ] is a
multiplication… isn’t it?
Students: Yes.
Teacher: But this [pointing to 5ðx 2Þ] is a single
number… because… didn’t we multiply both?
(Fig. 3)… We do the distribution only afterwards…
right?
The teacher continues the procedure of preparing the
equation to be solved in a simplified manner:
15ðx2Þ
6
¼ 2ðx2Þðx3Þ
2x6 . One student asks him when they
6
are going to multiply what is inside the parentheses.
2
How visual representations participate in algebra classes’ mathematical activity
Fig. 4 Gesture simulating a shower
11.
Teacher: Afterwards… when we are going to distribute we will do this… (Fig. 4).
The teacher writes a simpler expression 15ðx 2Þ ¼
2ðx 2Þðx 3Þ 2x2 and indicates the distributive law by
drawing curved arrows on the board:
In this episode, another activity system can be considered, with the following object: the association of the
parentheses with the shower in the product of algebraic
expressions. The subjects are the teacher and students.
Initially, the shower and parentheses could be considered
as two separate artifacts. However, along with this process,
the parentheses seem to be a cue to the use of the shower,
expanding the four ways of presentation mentioned in the
introduction of this paper.
From then onwards, we perceive that the association of
the visual image of the parentheses with the shower triggers a series of actions leading to the generalization of the
use of the shower to some situations to which it should not
be applied. For instance, Kleber makes it clear (line 7
above) when he suggests the use of the shower in the
expression 3ð5 ðx 2ÞÞ, which would result in
Other students also demonstrate the same understanding,
and the teacher tries to show that the distributive law is not
applicable in this situation (line 10 above), but he realizes that
the students are not with him, and he resumes the discussion.
The strong association of the shower with the visualization of the parentheses leads the students to apply the
101
distributive law to expressions with multiplication of three
factors, consisting of letters and numbers, associated by
parentheses, regardless of the operations indicated on the
expression, multiplying a number or letter by whatever is
inside the parentheses.
This triggers a tension in this activity, as two different
perspectives come into contact: the teacher’s use of the
distributive law and the students’ overgeneralized use of
the shower associated with the visual image of the
parentheses.
In summary, from the first to the second activity, we
notice a change in the activity object and a reconfiguration
of the shower artifact, through incorporation of another
visual sign (parentheses).
4.3 Activity 3: Overgeneralization of the use
of the shower in the product of algebraic
expressions
When the teacher realizes that many students are not correctly using the distributive law to carry out the procedures
for solving equations, he plans to discuss the matter.
However, differently from the previous activities, when the
subject was algebraic equations resolution, in this third
activity, the teacher changes the topic to focus on the
behavior of the operations. He explains to the students that
he has noticed they still make many errors and that he is
going to give an example to answer a question raised in the
previous class. To do this, he first goes back to an arithmetical expression in order to explore number properties.
He then writes on the board the double product (Fig. 5),
asking the students the result.
The students make different associations of the numbers
to compute the answer, and the teacher writes them on the
board. He explicitly asks if there are any doubts about the
task, and the students unanimously say they do not have
any problems understanding. To summarize this example,
the teacher calls the students’ attention to the fact that in all
the calculations involving the numbers 2, 3, and 5 there
were two multiplications and three factors. He then
Fig. 5 Back to arithmetic
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M. M. David et al.
Fig. 7 Scheme showing the factors
Fig. 6 Non-validity of the distributive law
rewrites the product 2 3 5 using parentheses, 2(3)(5),
in an attempt to show a resemblance with the pattern of the
algebraic expression 3 ð5 ðx 2ÞÞ, which appeared in the
previous class (Fig. 6). Then, he overuses the distributive
law in the product 2(3)(5) to persuade the students that this
leads to an error when dealing with algebraic expressions.
1.
Teacher: May I take the first one and go on distributing
it to the other two numbers?
Kleber: No… because they are not in the same… same
parentheses [the teacher does not seem to have listened
to the student’s argument].
Teacher: I cannot do this [meaning 2(3)(5) =
ð2 3Þð2 5Þ] here… may I?
Students: No.
Teacher: So… what happens? (if I do this)
Kleber: Because… (it) gives a wrong answer.
2.
3.
4.
5.
6.
(…)
8.
9.
10.
11.
12.
13.
14.
Teacher: So… may I (…) take 2… multiply and
distribute with respect to the other two (numbers)?
Students: No.
Kleber: No, because they are not in the same
parentheses.
Teacher: Ok?… So we may not do this… may we?
Teacher: So… when you see these numbers, does
anyone have any doubts?
Students: No.
Teacher: Does everybody agree?… So now it’s going
to be funny… Look here.
(…)
The teacher then writes the expression 7ðx 3Þðx 2Þ
on the board and asks how many factors are in the
expression. The students give various answers: 2, 3, 4, and
5. The teacher asks again how many factors are in the
numerical expression without parentheses. The students
answer three, and then the teacher draws parentheses
around every number, 2 ð3Þ ð5Þ, hoping that the students
will see an analogy between the two expressions: 2(3)(5)
and 7ðx 3Þðx 2Þ.
123
Fig. 8 Showing that each factor is a thing
17.
18.
19.
20.
Teacher: Actually, when we give a numerical value to
the x here ðx 3Þ… we will get another number…
When we give a numerical value to the x here
ðx 2Þ… we will get another number?… How many
numbers do I have to multiply here?
Students: Three.
Teacher: Three (factors)… 7… x minus 3… and
x minus 2.
Teacher: Is it clear… as it was there? So look…
The teacher writes a scheme on the board, writing how
many factors there are in the algebraic expression, what are
they, and how many operations there are (Fig. 7), emphasizing with gestures that inside each of the parentheses,
there is only one thing. He points out that some students are
still multiplying by 7 both the first thing ðx 3Þ and the
second thing ðx 2Þ.
26.
27.
28.
29.
Teacher: Now… So we do… 7x minus 21… but we
have made only one operation… haven’t we?… and
then we distributed… didn’t we? (Fig. 8).
Students: Yes.
Teacher: Done… Close the parentheses [showing
ð7x 21Þ]… We go on with another multiplication…
Isn’t it?
Students: Yes.
In this episode, we consider an activity system with the
object: when to apply the shower in the product of algebraic expressions. The shower maintains its influential role
as an artifact, and the teacher is the authority in the division
of labor.
As we can see, the teacher also highlights the use of
parentheses, but he does not emphasize which operations
are involved in the distributive law yet. At this point, the
shower is more stressed as an artifact than the property
How visual representations participate in algebra classes’ mathematical activity
itself. This approach seems to have reinforced the association that the students were making between the parentheses and the shower once the visualization of the
parentheses is used as a cue for the students to use the
distributive law. This association assigns more power to
this artifact in solving algebraic equations, when the
shower acquires agency impelling the students’ actions.
Looking back to the preceding activities, we notice that
the teacher gradually incorporates some characteristics of
the shower, influenced by the way the students deal with it.
We see this as a manifestation of the increasing influence
of the shower in the classroom’s activity, producing some
changes in the division of labor, as the teacher must share
his authority with the shower. Thus, the role played by the
shower moves from an artifact to a subject in the activity.
Latour (2005) argues that ‘‘any thing that does modify a
state of affairs by making a difference is an actor—or, if it
has no figuration yet, an actant’’ (p. 71). The shower, now
empowered by its agency, changes its role from an artifact
to a participant in the ongoing activity. The actions carried
out by this new subject (teacher with shower) serve as a
backdrop for the students’ actions.
We also may notice that tensions accumulate, as the
teacher remains focused on the similarities between structural properties of the real numbers and of algebraic
expressions, and the students are tied to the visual facets of
the shower, associated with procedures.
When the teacher uses the strategy of reverting to an
arithmetical example in order to explain the non-validity of
the distributive law with respect to multiplication, all the
students easily recognize that in the multiplication 2(3)(5)
there are three factors and two multiplications. The teacher
hopes that, after this, the students will make the transition
from the arithmetical example to the algebraic one
[7ðx 2Þðx 3Þ]. The symbol 2 3 may represent the
computational process of multiplying as a repeated addition ð3 þ 3Þ and the product ð2 3Þ is easily associated
with the number 6. The main argument used by the students to justify why the distributive law does not apply to
the multiplication 2(3)(5) was that it gives a wrong answer
(line 6). In the arithmetic context they could compare the
results using the concept of the product, which results in a
single-term answer.
An algebraic expression such as 7ðx 2Þ may also be
seen as a process: subtracting 2 from x and then multiplying the result by 7; or, structurally, as an object that
represents a certain number. However, for the students to
conclude that there are also two multiplications and three
factors in 7ðx 2Þðx 3Þ, they would have to see the
algebraic expressions ðx 2Þ and ðx 3Þ structurally, as
objects, instead of operationally, as processes. But, for
them, the expression ðx 2Þ is not a number, since the
subtraction cannot be finished.
103
Many researchers, such as Gray and Tall (1994) and
Sfard and Linchevski (1994), point out that students have
difficulties distinguishing objects and processes. According
to Sfard and Linchevski (1994) the duality process–object
is related with different ways of thinking—operational and
structural. In mathematics, operational conception usually
precedes the structural one; what is conceived as a process
at one level becomes an object at a higher level. So, when
working with algebraic symbols what one actually sees in
them ‘‘depends on what one is prepared to notice and able
to perceive’’ (p. 192).
According to Sfard (1991), the transition from computational operations to abstract objects is a long and inherently difficult process, as illustrated by the previous
classroom episode. Therefore, the lack of immediate success in the teacher’s attempt to explain the distributive law
in the algebraic context going back to the numerical one is
not surprising.
The tension already perceived in the last activity is not
resolved, and it evolves into the use of a visual representation for a specific procedure as if it were a generic representation applicable to all situations that combine
parentheses and operations.
This calls for a deeper discussion about the role of the
visual display in teaching algebraic procedures in mathematics classrooms. This tension has also been analyzed by
literature in the field, taking different perspectives. Russell
et al. (2011) argue that many middle school algebra teachers
note that students misuse the distributive property. They
claim that, probably, when the students extend this property
to multiplication, they see a resemblance in the pattern of
the symbols to the ones of the distributive law. In this case,
they do not recognize an application of the associative law.
Therefore, the authors propose that teachers should present
their students with descriptions of the properties of the
operations when doing arithmetic. For instance, when
working with specific examples to explore the commutative
law, the teacher may take the regularity present in those
examples as an explicit focus of investigation, leading the
students to think in terms of generalization, asking them ‘‘to
think about whether changing the order of the addends
maintains the sum only for specific cases or whether it is
true more generally and to explain how they knew’’ (Russell
et al. 2011, p. 48). In the case reported, since the students
were joining two sets of cubes to illustrate addition,
switching positions of the sets did not change the number of
cubes. Moreover, this cubes model could represent any pair
of numbers, leading the students to conclude that the sum
would always be the same since ‘‘You are not adding
anything or taking anything away’’ (p. 47).
These authors conclude that a strong foundation in
whole number computation can be extended to algebraic
symbols, providing crucial links between arithmetic and
123
104
M. M. David et al.
13.
14.
15.
16.
17.
18.
19.
20.
Fig. 9 Making associations
algebra, but they do not make any specific connections
between the overgeneralization processes they analyze
with any visual components possibly involved.
4.4 Activity 4: Connecting the shower
with the distributive law
In this episode, the teacher’s aim was to review some
central issues related to fractional algebraic equations. He
proposes several exercises requiring the use of the distributive law, but he does not mention anything related to
the distributive law and/or the shower.
Occasionally, some students ask for the teacher’s help,
and he seems to notice that the students still have doubts
related to the use of the shower/distributive law. He goes
to the blackboard and writes the algebraic expression
3 ðx þ 5Þ ðx 5Þ to review the use of the distributive law.
Next, he explicitly points out the operations involved, and
he states clearly that they are applying the distributive
law:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Teacher: Look here [pointing to 3 ðx þ 5Þ ðx 5Þ]…
Everybody… How many multiplications do I have
here?
Students: Three…
One student: Two…
Students: Three.
Teacher: Laura…
Laura: I think there are two… 3 times one number
and the (resulting) number times the other…
Teacher: This [pointing to (x ? 5)] here… is a single
number… isn’t it?
Student: Yes.
Teacher: It’s inside the parentheses… and…
although there is an addition there… it represents a
number… doesn’t it?
Teacher: So… it represents a number, and we
make… first… this [pointing to 3(x ? 5)] here…
and we find out how much it is… (Fig. 9)
Teacher: Then I ask… how do I solve this?
Student: No matter the order…
123
21.
22.
Another student: Distributive…
Teacher:… Do I solve this here?
Students: Shower…
Teacher: Shower… 3 times x is going to be…
Students: 3x.
Teacher: 3 times 5…
Students: 15.
Teacher: Done… It’s 3x þ 15… Now we put parentheses here [pointing to 3x þ 15] not to be confused… and we multiply by the rest… the second
multiplication… and then I make this [pointing to
ð3x þ 15Þðx 5Þ] here… There is another shower…
and I solve it.
Teacher: Can I make this multiplication [pointing to
ðx þ 5Þ ðx 5Þ] here and then multiply by 3?
Students: You can.
(…)
23.
24.
25.
26.
Teacher: Isn’t it?… How many multiplications do we
have there?
Student: Two.
Teacher: Two, isn’t it? Sometimes one may think…
two multiplications… No… because if you have to
make here and here [showing with his hands curved
arrows linking 3 to x and 3 to 5]… but this is a
property of the multiplication isn’t it?
Teacher:… Understood?… Check this in the exercises because there are a lot of people who are
confused about this…
At the end of this class, the teacher calls Laura to present
her solution for the fractional equation. When she gets to
the expression ð2xÞðx 3Þ þ 3 ðx 1Þ he asks her to tell
the answer. The student answers, and the teacher interprets
what the student did to arrive to this answer, making an
association between the shower and the distributive law:
‘‘You made… the shower here… the distributive… didn’t
you?’’
In this last episode, we consider an activity system
whose subject is the teacher and the students, and whose
object is the connection of the shower with the distributive
law. The shower returns to the role of an artifact, but now
in an empowered position, since its use was accepted by the
teacher, and the students start using it still in a procedural,
but less mechanical, way. There is a movement
approaching the use of the shower made by the students
and the teacher.
The teacher’s and students’ actions develop in such way
that the evolving tension between the teacher’s use of the
distributive law and the students’ overgeneralized use of the
shower associated with the visual image of the parentheses
seems to be momentarily attenuated in this last activity.
This tension diminishes because all the different ways of
How visual representations participate in algebra classes’ mathematical activity
presentation associated with the shower, including those
privileged by the students (parentheses, curved arrows,
words), are gradually incorporated into the teacher’s
explanations, meaning that he accepts the shower as one
more artifact for teaching the procedures leading to the
correct application of the distributive law in algebraic
activities.
4.5 The activity system: The use of the shower
in the product of algebraic expressions
Given the complexity of the activity system composed by
all activities on the use of the distributive law in the
product of algebraic expressions, we can use different
lenses in our analysis (zooming in and out, inward and
outwards), at times focusing on short-term classroom
activities, alternating with moments when we take a more
distant view over longer-term systems of interconnected
activities. Looking retrospectively to the historicity of this
activity system, we perceive that in the second and third
activities, the teacher tries to explain the use of the shower,
but he does not emphasize the operations involved. Only in
the fourth activity does he make a stronger association
between the mathematical facet and the other facets of the
shower.
By zooming out on the constellation of the four activities
considered, we gain a clearer perception of the mobility
within the stability of the overall activity system—the use
of the shower in the product of algebraic expressions. The
object goes through some fluctuations, but there is stability
in that the objects of the four activities are all associated
with the use of the shower in the product of algebraic
expressions. The subjects and community are quite stable,
despite the fact that the shower momentarily acquires some
agency. The artifact shower momentarily plays a central
role in the activity, acquiring agency, moving towards the
position of a subject in the activity. The rules become more
flexible as the students’ use of the shower approaches the
teacher’s use, indicating a closer connection between the
mathematical object (distributive law) and the procedures
performed by the students. The students gradually come to
share an authority position with the teacher, establishing a
more horizontal division of labor in the classroom.
Although the teacher never declares the use of the
shower as a practice that can be accepted by school
mathematics, he himself starts using the artifact privileged
by the students in order to establish a dialogue with them
on a more horizontal basis. The zoom out procedure we use
allows us to consider the four activities as actions of the
activity system—the use of the shower in the product of
algebraic expressions—and the movement of changing
lenses enables us to build an overall understanding of the
activity system.
105
5 Concluding remarks
As shown by our analysis, the classroom activity changed,
impelled by the use of the visual display (little shower)
associated with the distributive law. According to the
theoretical framework adopted we went beyond the relation
of the subjects (teacher and students) with the mathematical concepts, intermediated by visual artifacts, and we have
considered other relations by the incorporation of different
components of the classroom activity, such as the rules,
community, and division of labor. For example, the
shower, initially seen as a mediating artifact to represent a
sequence of coordinated actions to facilitate the application
of the distributivity, became more empowered along with
the activity, gaining agency and more visibility in the
classroom.
The perspective adopted also offered the possibility to
capture the historicity of the classroom activity illuminating tensions between different ways of visualizing the
distributive law. Radford (2006) explains that in order to
make a mathematical property apparent, ‘‘students and
teachers make recourse to signs and artifacts of different
sorts (mathematical symbols, graphs, words, calculators
and so on)’’ (p. 6) in a process of objectification. In our
case, in an attempt to make the distributive law apparent
for the students, the shower takes a privileged position, but,
while the teacher sees primarily the operations involved in
this law, the students see, first of all, the parentheses,
irrespective of the operations involved, and then the curved
arrows of the shower.
For the teacher, algebraic and numerical expressions
behave the same way and, irrespective of what they represent, the most important thing is that they can be multiplied, added, subtracted, according to the properties of the
algebraic structures they belong to. For the students,
algebraic expressions represent actions to be performed
rather than elements of a given algebraic structure. We
believe that, latent behind the tensions perceived, there is a
contradiction between the structural way the teacher signifies the distributive law (originating from his mathematical education) and the procedural way the students
signify it (originating from the strong visual appeal of the
shower display).
In the teaching of algebra, visualization may be geared
(by the teacher, the students or the visual displays) towards
the properties of algebraic objects and/or to the procedures.
Some visual displays can be powerful in helping students
to visualize certain algebraic notions. But if they emphasize only the procedural aspect, as the shower, it is not
surprising that many students tend to overuse them in a
mechanical way in other situations with a visual resemblance to the ones to which they can be applied. That is,
there may be pros and cons in the use of some visual
123
106
displays in algebra classes, and we argue that it is very
important for the teacher to be aware of this if s/he is to
benefit from the use of such displays.
Contrary to his general disposition of not allowing a
mechanical use of tricks, this teacher succeeded in attenuating the tensions perceived in the classroom, when he
momentarily extended the classroom’s repertoire related to
the distributive law, including the gestures, curved arrows,
and the word shower. He was led to discuss these other
facets of the shower as a result of the students’ insistence
on using them, given the widespread use of this visual
display in the community.
This is a typical example of the experiential knowledge
(Tardiff 2008) of mathematics teachers. It is a practical form
of knowledge that is not taken into consideration by the
school’s curriculum nor by mathematics teachers’ pre-service education.5 The use of the shower to teach the distributive law is a trick that is validated through teachers’
everyday work, becoming an inseparable part of the distributive law for the students. As this kind of knowledge is
not systematized, although teachers seem to be usually aware
of the strong appeal of this sort of display, they may not
foresee all the implications of its use, as, for instance, the
possibility of the visual dimension overlaying and hiding the
meaning of the mathematical ideas for the students.
We believe that the teacher we observed, in spite of his
short teaching experience, was attentive enough to perceive
his students’ overuse of the shower and succeeded in
guiding at least some of them to understand that they
should not disconnect the use of this display from the
mathematical meaning it was introduced to represent.
Thus, we learned from this teacher how, instead of losing
the great potential for visualization offered by these displays by avoiding them or prohibiting their utilization by
the students, the teacher should face the tensions they
sometimes create with the teacher’s conceptions and discuss in an open way when and how they should be applied.
Acknowledgments The authors want to declare, first of all, their
gratitude to the teacher and students involved in this study, for all we
have learned with them. We also wish to thank the Conselho Nacional
de Desenvolvimento Cientı́fico e Tecnológico—CNPq, Fundação de
Amparo à Pesquisa de Minas Gerais—FAPEMIG, and Pró-reitoria de
Pesquisa da Universidade Federal de Minas Gerais—PRPq/UFMG for
the financial support received.
References
Arcavi, A. (2003). The role of visual representations in the learning of
mathematics. Educational Studies in Mathematics, 52, 215–241.
5
It can, however, be found in a quick Google search on several sites
that present materials for teachers and/or solved problems for the
students.
123
M. M. David et al.
Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation
in the mathematics classroom: Artifacts and signs after a
vygotskian perspective. In L. English, M. Bartolini Bussi,
G.Jones, R. Lesh, and D. Tirosh. (Eds.), Handbook of International Research in Mathematics Education. (pp. 746–805).
Second revised edition. Lawrence Erlbaum, Mahwah, NJ.
Bishop, A. (1991). Mathematical enculturation: A cultural perspective on Mathematics Education. Dordretcht: Kluwer Academic
Publishers.
Brown, K., & Cole, M. (2000). Socially shared cognition: System
design and the organization of collaborative research. In D.H.
Jonassen and S.M. Land (Eds.), Theoretical Foundations of
Learning Environments. Lawrence Erlbaum, p. 197–214.
Casselman, B. (2000). Pictures and proofs. Notices of the AMS,
47(10), 1257–1266.
David, M. M., & Tomaz, V. S. (2012). The role of visual
representations for structuring classroom mathematical activity.
Educational Studies in Mathematics, 80, 413–431.
Duval, R. (2006). A cognitive analysis of problems of comprehension
in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.
Engeström, Y. (1987). Learning by expanding: An activity-theoretical
approach to developmental research. Helsinki, Finland: OrientaKonsultit.
Engeström, Y. (2001). Expansive learning at work: Toward an
activity theoretical reconceptualization. Journal of Education
and Work, 14(1), 133–156.
Engeström, Y., & Sannino, A. (2010). Studies of expansive learning:
Foundations, findings and future challenges. Educational
Research Review, 5, 1–24.
Fischbein, E. (1993). The theory of figural concepts. Educational
Studies in Mathematics, 24, 139–162.
Freitas, E., & Sinclair, N. (2012). Diagram, gestures, agency:
Theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics, 80, 133–152.
Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: A
proceptual view of simple arithmetic. Journal for Research in
Mathematics Education, 26(2), 115–141.
Jonassen, D.H. (2000) Revisiting Activity Theory as a framework for
designing student-centered learning environments. In D.H.
Jonassen and S.M. Land (Eds.), Theoretical Foundations of
Learning Environments. Lawrence Erlbaum, p. 89–120.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in
schoolchildren. Chicago: The University of Chicago.
Latour, B. (2005). Reassembling the social. An Introduction to actornetwork-theory. New York: Oxford University Press Inc.
Leont’ev, A. N. (1978). Activity, consciousness, personality. Englewood Cliffs, New Jersey: Prentice-Hall.
Mason, J., Stephens, M., & Watson, A. (2009). Appreciating
mathematical structure for all. Mathematics Education Research
Journal, 21(2), 10–32.
Mesquita, A. (1998). On conceptual obstacles linked with external
representation in geometry. Journal of Mathematical Behavior,
17(2), 183–195.
Núñez, R. (2006). Do real numbers really move? Language, thought,
and gesture: The embodied cognitive foundations of mathematics. In R. Hersh (Ed.), 18 Unconventional essays on the nature of
mathematics (pp. 160–181). New York: Springer.
Parzysz, B. (1988). ‘‘Knowing’’ vs ‘‘seeing’’. Problems of the plane
representation of space geometry figures. Educational Studies in
Mathematics, 19, 79–92.
Presmeg, N. (2006). Research on visualization in learning and
teaching Mathematics: Emergence from psychology. Introduction. In A. Gutierrez, P. Boero (Ed.), Handbook of research on
the psychology of Mathematics: past, present and future. (pp.
205–236). Rotterdam: Sense publishers.
How visual representations participate in algebra classes’ mathematical activity
Radford, L. (2003). Gestures, speech, and the sprouting of signs: A
semiotic-cultural approach to student’s types of generalization.
Mathematical Thinking and Learning, 5(1), 37–70.
Radford, L. (2006). Algebraic thinking and the generalization of
patterns: A semiotic perspective. In S. Alatorre, J. Cortina, M.
Sáiz, and A. Méndez (Eds.), Proc. 28th Conf. of the Int. Group
for the Psychology of Mathematics Education (Vol. 1, pp. 2–21).
Mérida, México: PME.
Rivera, F. (2011). Toward a visually-oriented school mathematics
curriculum. New York: Springer.
Russell, S., Schifter, D., & Bastable, V. (2011). Developing algebraic
thinking in the context of arithmetic. In J. Cai & E. Knuth (Eds.),
Early algebraization: A global dialogue from multiple perspectives (pp. 43–69). Berlin: Springer.
Sfard, A. (1991). On the dual nature of mathematics conceptions:
Reflections on processes and objects as different sides of the
same coin. Educational Studies in Mathematics, 22, 1–36.
107
Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of
reification: The case of algebra. Educational Studies in Mathematics, 26, 191–228.
Sinclair, N., & Tabaghi, S. G. (2010). Drawing space: Mathematicians’ kinetic conceptions of eigenvectors. Educational Studies
in Mathematics, 74, 223–240.
Tardiff, M. (2008). Saberes Docentes e Formação Profissional
[Teachers knowledge and Professional Education] (9th ed.).
Petrópolis, R.J: Vozes.
Tomaz, V. S., & David, M. M. (2011). Classroom activity promoting
students’ learning about the use of drawings in geometry. In B.
Ubuz (Ed.), Proc. 35th Conf. of the Int. Group for the
Psychology of Mathematics Education (Vol. 4, pp. 257–264).
Ankara, Turkey: PME.
Zimmermann, W., & Cunningham, S. (Ed.). (1991). Visualization in
teaching and learning Mathematics. Mathematical Association
of America MAA Notes, 19.
123