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MATHEMATICAL KNOWLEDGE AND INDIVIDUAL EXPERIENCE
A STUDY OF CHILDREN'S MATHEMATICAL LEARNING
WITH THIRD GRADERS IN BRAZIL

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Maria Bellini Alves Monteiro

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MATHEMATICAL KNOWLEDGE AND INDIVIDUAL EXPERIENCE:
A STUDY OF CHILDREN'S MATHEMATICAL LEARNING
WITH THIRD GRADERS IN BRAZIL

By

Maria Bellini Alves Monteiro

ADISSERTATION

Submitted to
Michigan S tate University
in Partial Fulfillment of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

Department ofTeac her Education

1994

ABSTRACT

MATHEMATICAL KNOWLEDGE AND INDIVIDUAL EXPERIENCE:
A STUDY OF CHILDREN'S MATHEMATICAL LEARNING
WITH THIRD GRADERS IN BRAZIL

By

MariaBellini Alves Monteiro

This is a study about third grade students’ mathematical reasoning both in mathematics
classes and outside school. For three months, I worked with 30, 10-year-old children attending
a school in a neighborhood of unsldlled workers. I spent the time performing participant
observation in the school, classroom and in the outside community--in stores, bakeries, butcher
shops, as well as in the streets.

The students were asked to solve division problems during informal interviews and in the
classroom context at the end of the period we had spent together. The problems were
elaborated, based upon situations occurring in their daily experiences. They worked with
eight, word problems involving division, at different levels of difficulty.

I found a variety of strategies those children used to solve division problems out of the
school. Those strategies show the complexity of mathematical reasoning they were able to do.
Out of school, students solved division problems by rounding numbers before operating them
and they did estimation of possible results. The students also simplified calculations by
doing successive additions and subtractions. They used some proportional reasoning and
they worked with the remainder to balance the division. However, in their mathematic's
class, their mathematical reasoning was limited to formal algorithms in solving the same
problems. One important consequence of this difference in mathematical reasoning was
their success in achieving a correct answer which was much higher in the out-of-school

setting.

To conclude, I explored the differences between the strategies students used in
and out of school to highlight the possible existence of two distinct views of
mathematical knowledge. In doing that, I studied the roles of particular aspects of the
students' social and cultural context in the construction of the notion of division. I
identified the following three connections between the students' idea of division and
some of the students' points of view on their cultural and social context: 1) division as
separation of equal quantities, 2) division as an act of sharing, distributing and giving,
and 3) division in a physical sense, that is splitting in two or more pieces or groups.
Finally, based on my findings I have included some possible implications for the practice
of mathematical education based on my findings, especially related with mathematical

knowledge we teach and how we teach it.

COPyn'ghtby
MARIA BELLINI ALVES MONTEIRO
1994

To my mother

ACKNOWLEDGMENTS

This dissertation was possible thanks to:

The Brazilian Government through CAPES which granted a scholarship.

UFJF, the Brazilian University, which granted me a leave of absence during my course
work.

The group of students and teachers I worked with in Brazil to collect the data for this
dissertation.

Drs. Ann Millard, Sandy Wilcox, Ed. Smith and Douglas Campbell, the members of my
Guidance Committee, for the support they gave me throughout the time we worked together.

Dr. Everson Alves Miranda, my dear friend, who shared with me the difficultdecisions I
had to make.

Dr. James Gallagher, my advisor and the primary person responsible formy achievement
of the final step of the Doctoral Program. He made me believe it was possible.

vi

TABLE OF CONTENTS

Chapter Page
I. INTRODUCTION ............................................................................ 1
Plan of the Dissertation ....................................................................... 5
II. THEPROBLEM ............................................................................... 7
Introduction ....................................................................................... 7
Conceptual Framework forAnalyzin g
Mathematical Teaching and Learning ............................................. 8
Mathematical Content Knowledge and Context .............................. 15
Selected Readings on Students Doing Division ............................... 21
III. CHILDREN’S PERSPECTIVES ..................................................... 29
Introduction ...................................................................................... 29

Children’s Perspectives About Their
Social Milieu .................................................................................... 31
Students' Perspectives About Mathematics. .................................... 39

IV. CONTEXT 8 OF CHILDREN’S EXPERIENCES

INVOLVING MATHEMATICS ..................................................... 47
Introduction ...................................................................................... 47
The School Context ......................................................................... 49
The Classroom Context ................................................................... 53
A Class About Division ................................................................... 57
Mathematics in Everyday Life ......................................................... 67
V. CHILDREN DOING MATHEMATICS. ........................................ 84
Introduction ...................................................................................... 84
Solving Problems in the Classroom Context ................................... 87
Solving Problems Outside the Classroom Context .......................... 96

vii

VI

CONCLUSIONS .............................................................................. 122

Are There Two Different Mathematics

in Two Different Worlds? ................................................................ 122
Constructing the Notion of Division ............................................... 132
Reflections on the Students' Ideas of Division ................................. 138

Implications for Practice and Recommendations for

Further Research .............................................................................. 141
APPENDICES

Appendix A ..................................................................................... 145
Appendix B ..................................................................................... 146
BIBLIOGRAPHY ............................................................................ 147

viii

Table
1
2
3

LIST OF TABLES

Results of Students Solving Problems in the Classroom ................. 128
Results of Student Solving Problems in the Real World Context.... 129
Results of Students Solving Problems ............................................. 129

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LIST OF FIGURES

Studentcrying .................................................................................... 1
Power relationships student perceive in society ............................... 35
Mathematics class ............................................................................. 44
Balloons ....................................................................................... 45
A set ....................................................................................... 60
A set and subsets .............................................................................. 61
A set of tampinhas ............................................................................ 62
Dividing tampinhas ......................................................................... 62
Playing Field for Queimada ............................................................. 69
Playing bete 1 .................................................................................. 70
Playing here 2 .................................................................................. 71
Playing here 3 ................................................................................... 71
Playing flag ..................................................................................... 72
Playing arco. ................................................................................... 74
Playing school ................................................................................. 75

CHAPTER I

INTRODUCTION

I developed this research with 30 third graders in an elementary school in Brazil.
In its beginning two interesting events happened. I was exchanging journals with the
students as a way of communicating and commenting on their experiences involving
mathematics inside and outside school. When I asked them to show me how they had
mathematics in their games, homes, or activities outside school, Francisca drew the

picture that follows:

 

 

 

Figure 1 Student crying

On another day gathering data about the students’ perspectives of their social

milieu, Rosa, reacting to the picture showing the Brazilian President, said:

If I could talk to the president...

I was going to ask him to build more hospitals, more houses for the homeless... to

stop increasing the prices of materials used in building houses. But he could do

more. He could stop increasing the prices for rice, beans, sugar and the milk... He
could provide more opportunities for employment for the homeless beggars, be-
cause they perhaps wanted to have one and cannot find it even though they look for
it. They, the beggars ask him for food, but what is the point of giving food for one
day? All the other days they will remain hungry because it won’t be enough for
every one of them. So, it is better to lower prices of things and provide for more
employment opportunities.

This quote and the picture reflect two different sides of the students’ way of
approaching mathematics. In the picture, the child’s tears seem to show her frustration
and sadness while she is doing her mathematic work. The picture also shows the child
doing calculations that are usually the kind of mathematics the student has at school.

On the other hand, the quote reveals a different perspective toward mathematics: It
is the real situation where the child lives. Rosa is expressing the idea of division related
to sharing and fairness in a context where not only quantitative relationships were
considered

The picture and the quote reveal a paradox between the world of school
mathematics and the world of the students’ experiences related to their social and cultural
milieu. That is the paradox I will be addressing in this dissertation.

The study site was a third- grade elementary school, mathematics class in the
southeast part of Brazil. I worked with a group of 30 students, ten years old on average,
who lived in a neighborhood of unskilled workers in a medium-sized town.

I began by observing students in mathematics classes. As 1 did so, I was looking
for components of the student’s daily experience surfacing during the process of

classroom mathematical learning. I believed that events from the students' everyday life,

when brought into the situation of mathematics class, would reveal how both school
mathematics and the students' social and cultural background interrelate in the learning
process. In my second visit to the students in the classroom, I proposed to them to keep
a journal with a general note saying that it would be the instrument they could use to tell
me more individually about their experience with mathematics. They could write, make
drawings, use pictures, or do anything else of their concern to establish the conversation
with me. We exchanged these journals during the whole observation period. The period
of observation consisted of three months of daily visits to the classroom and the school
for at least three hours each day. Besides observing class, I interviewed the teacher
responsible for the third grade with whom I was workingone hour every week . In
addition, we were together every day during lunch and break time where students were
around in informal conversations. At school, I could interview the principal, supervisor,
counselor, other teachers responsible for the library and gymnastics, and the secretary.
My concern was to have a comprehensive view of the school life that in some way could
affect students' learning in the classroom situation.

What I wrote in my personal journal after the first month of observations was a
reflection of my perplexities. I did not perceive a clear manifestation of the encounter
between school mathematics and the students’ experiences outside school in the
construction of mathematics learning. At that time I wrote:

Why is it? What can I do to see better? Am I looking in a wrong direction?

What should I look for? If I am wrong and there is nothing there to see, I have to

doubt whether students construct knowledge in the formal process of schooling. If

they construct knowledge in the learning situation, what takes part in this construc-

tion? (Personal notes. Journal in September, 1990)

At that time I began interviewing students out of the classroom context. We talked
freely about their everyday experiences in a variety of situations involving mathematics.

Daily problems they faced came out easily, as well as their own ways of solving them.

Thus, I realized that I should pursue this side of data collection more deeply. I went to
each store, grocery, bakery and butcher’s shop near where students lived and I talked to
the vendors and observed children buying and making change by themselves and with
their parents. In addition, also observed children playing games on the streets. I talked to
them about their activities involving mathematics and their ways of solving problems
they face in their daily activities. Finally, I went to many of the childrens’ homes to visit
their parents. Those contacts were very rich and helped me to understand the children’s
cultural and social background.

Staying in the community was essential to uncover relevant themes that were part of
the students’ daily lives. I also felt that I had to look again more carefully at the way
children were doing mathematics in the classroom, in contrast to what they were doing
outside school. To do that, I planned an interview with a group of between two and four
students, to explore more deeply two basic points: 1) the students’ perception of their
social and cultural milieu and 2) their way to solve mathematics problems involving
division. Students were studying division in their mathematics classes and, thus, that
topic became the focus of the research. In my last day of observation, I asked the
children to solve the same problems in the classroom that they had solved in the
interview. I did it in the same way the teacher used to do it with word problems in the
classroom situation.

The results I found helped me to look at the paradox I mentioned in the beginning
of this Introduction and to learn from it. The report of this dissertation is my account of
interpreting this paradox and trying to illuminate the question that remains about the role
that culture and social factors play in how students construct mathematical knowledge in

elementary school.

E] E 1 12° .

I will report this research starting with the problem I focused on and a description of
four propositions that form the conceptual bases I used to analyze mathematical teaching
and learning in this study. That framework guided my readings about mathematical
content knowledge and context. Chapter One is a summary of those readings and a
literature review about students’ way of doing division.

In Chapter Two, I describe the children’s perspectives about their social milieu and
about mathematics. They are important to help us to understand children’s ways of doing
mathematics and how they relate their own experience within the school and in the
community as a whole.

In Chapter Three, I explore children’s experiences involving mathematics in two
contexts. First I describe the formal situation of the classroom. The class about division
shows how the teacher approaches the theme and how children react to the teaching
situation. On the other hand, many situations that students have to face in their everyday
life require mathematical reasoning. I describe some of these situations such as childrens
buying things in stores, bakeries and in butcher shops, and children helping their parents
at home or playing games.

In Chapter Four, I analyze how those children deal with mathematics, solving word
problems. I will distinguish two contexts in doing the analysis: the classroom context
and the outside-of-the-school context. The mathematical content involved in the situation
problem was division. I took both contexts, in and out of the school, into consideration to
analyze how children approach the problems and reason to solve them. I found important
differences in the students' mathematical reasoning in dealing with word problems in the
two situations.

Finally, in Chapter Five, I discuss the possibility of the existence of two different
ways of doing mathematics in school and outside of school. To conclude this study, I

consider the construction of the notion of division. I present three connections between

the students’ perspectives of their social milieu and how they make sense out of the

idea of division in mathematics.

CHAPTER II

THE PROBLEM

Inmdusztitm

This is a report about a study done in Brazil with 30, third graders about
mathematical learning and its context. The study is a result of my reflections during my
doctoral studies, since the fall of 1987, about cultural and social issues related to the
educational process. During this time, I became interested in Studying mathematical
learning and social and cultural context. An approach that contextualizes mathematical
learning has been the focus of attention of many researchers lately (Lerman, 1989;
Bishop, 1988; Lave, 1984; Carraher, 1988). Drawing from this literature I have defined
the problem in this study in terms of looking at the students’ mathematical reasoning both
in and outside the school context. I did it to find out how cultural and social components
come together with mathematical knowledge as it is taught within the school, and to
contrast students' use of mathematical knowledge in the two contexts. This chapter
illustrates how this problem evolved from the very beginning until now, based on the
literature I reviewed while looking for theoretical support to my questions.

My first approach was to look at how context has been focused on in the studies of
teaching and learning mathematics. I became familiar with current research in
mathematics education focusing on teachers’ and students’ understanding about
mathematical knowledge and the culture of the classroom. From this research I learned
that different approaches to mathematics shape different ways of teaching, with
consequent implications for how students approach mathematical learning situations.
The basic issue is the construction of meanings.

Then, I became puzzled about the idea of students constructing their own

understanding of mathematical knowledge and the role of cultural and social bias on this

7

construction. Mathematical knowledge as a cultural phenomenon and its implication for
the learning and teaching of mathematics is a concern within the field of
ethnomathematics (Gerdes, 1988; D’Ambrosio, 1985; Ferreira, 1988). Mathematical
knowledge as a cultural phenomenon constitutes the compelling ground upon which I
based my research. My study was an initial approach to the role played by cultural
context in how a group of students constructed their mathematical knowledge. Thus,
studies of mathematical learning in different contexts were the main source in refining
the basic questions for this study.

The following are the questions in this research:

1. How are students’ daily experiences taken into account in mathematics class-

rooms?

2. How do components of students daily experiences surface during the process of

classroom mathematical learning?

3. Which contrasts, conflicts, and confrontations emerge from mathematics teach-

ing in a school setting?

4. How do individuals encounter mathematical knowledge in a classroom setting?

5. What role do cultural and social factors play in how students construct math-

ematical knowledge in elementary school?

Next, I will explain the conceptual framework which provides the basis of my

understanding of mathematical teaching and learning.

01 '0 _1_ -_ 'oowrkf ‘ n1 " M 721. T}. in . c ‘a._un.'
From the studies I reviewed on mathematics and mathematics education, four
propositions were derived.
mil—9923 Mathematical concepts are not “pure concepts” or “absolute
truths” about physical and quantitative relations in the experiential world.

I will discuss proposition one by examining contributions both from the history of

mathematics and from the philosophical debate on the objective existence of
mathematical abstractions. A full discussion of the arguments involved is impossible
within the limits of this dissertation. Hence my purpose is to highlight some thoughts
fiom scholars who have discussed this matter to shed light on the proposition. By doing
that I will also outline the aspects of the proposition that I took into consideration.

Birnbaum (1987) conducted an extensive discussion of what mathematics is. In
doing, that he analyzed different traditions which respond differently to the question of
what mathematics is. From those traditions he concluded that what varies and makes
them different is the nature of the answers regarding the acceptance or denial of the
objective existence of mathematical abstractions. The question of what mathematics is
moves towards the relation between mathematics and reality. Does mathematics describe
the real world as it is? Does mathematics refer to “absolute truths” about the physical,
quantitative relations in the universe? Is mathematics discovery or creation?

Machado (1987) examined the relationship between mathematics and reality. He
took three, contemporaneous traditions (logicism, formalism, and intuitionism) to study
that relationship. In his search for the relationship between those three traditions, he
posited the argument that mathematics is embedded in the socio-historic process where
it is produced. He presented his search for a way of looking at mathematics in the

following terms:

Uma visa'o que explicite a situacdo da materndtica como objeto da cultura,
como ferramenta de trabalho, que revele com clareza o quanto a matemdtica
esta inserida no processo histo’rico-social onde é produzida e que ela ajuda a
produzir. Uma visa'o que logre a superaca'o do mito da matemdtica
hermética, ciéncia dos “eleitos” cujafunca'o primordial, como a de outros
mitos, e a justificacdo de privilégios de diferentes ordens através do elogio da
técnica, ou de uma dimensao dela (Machado, 1987, pp. 16-17). (We shall
search for a vision that makes explicit mathematics as an object of culture, a
working tool, that clearly reveals how much it is inserted in the socio-histon'c
process where it is produced. That vision should overcome the myth of an
hermetic mathematics, science of the few “electe ” whose main function, as it
is the case of other myths, is the justification of privileges of various levels by

10

overvaluing technique as a whole or one of its dimensions).

Among intuitionists, logicists, and forrnalists, there is agreement over the idea that
mathematics is manipulation of symbols in accordance with certain rules. Intuitionists
differ from the other two by viewing mathematics as intuitive constructions of the formal
manipulations of symbols. While logicists rely on refined logic to establish foundations
for mathematics, intuitionists believe that the truth can be directly intuited (Kline, 1980).

Machado (1987) traced the formalist u'adition back to Kant. According to
Machado, forrnalists start with basic truth, from which they derive axioms. Using rules
of inference, theorems are defined from the axioms. Thus, theorems are demonstrable
elements of truth.

What all three traditions hold in common is the denial of objective existence of any
mathematical reality external to the mathematician. This idealistic position considers
mathematical concepts and theories as pure products of the mind with no relation to
reality (Morozov, 1987).

Kline (1962) discussed the same question of what mathematics is in terms of the
ontological status of its fundamental objects. In other words, the question is whether
concepts, axioms, and theorems exist in some way in the objective world or whether they
are entirely the result of human creation? He discussed these and other questions by
using a cultural approach to the value of mathematics.

Kline’s account of the non-Euclidian geometries is one example of his approach
(Kline, 1962). He pointed out that the existence of the two geometries provided a
significant moment in the construction of the mathematical body of knowledge. This
was a moment of realization that mathematics does not offer definitive truth.
Furthermore, he asserted that mathematical laws are “merely approximate descriptions
and, however accurate, no more than men’s way of understanding and viewing” (Kline,

1962 p. 576). Kline also discussed about the nature and values of mathematics. He

11

stated:

In those domains where it [mathematics] is effective it is all we have; if it is
not reality itself, it is the closest to reality we can get. Mathematics then is a
formidable and bold bridge between ourselves and the external world.
Though it is a purely human creation, the access it has given us to some
domains of nature enables us to progress far beyond all expectations. Indeed
it is paradoxical that abstractions so remote from reality would achieve so
much. Artificial the mathematical account may be, a fairy tale perhaps, but
one with a moral (Kline, 1962, p. 676).
The argument that mathematics is a human creation is reaffirmed in a discussion about
the loss of certainty in the mathematical domain (Kline, 1990).

Reflecting on the literature I have presented so far, one may draw a conclusion that
there is an objective, experiential world out there and that mathematical knowledge is a
result of human creation dealing with that world. Thus, one may assume that
mathematics does not hold “absolute u'uths” about physical and quantitative relations in
the experiential world. As a consequence of that assumption, mathematics may be
subjected to revisions and modifications according to different and new historical
developments undergone by human beings. Also, one may state that mathematics is
deeply embedded in the human culture. These conclusions bring me to the second
proposition.

Emmsitjon ng: Mathematical knowledge is culturally bounded.

This proposition is actually an extension of the first one. If it is assumed that
mathematics is created, not discovered, and it is the result of human enterprise, the
consequence is to admit that the very same cultural conditions of human existence are to
be found in the mathematical knowledge humanity is creating. To state it in other terms,
if people exist within a time and space frame that provides the conditions of their
experience, then, all that they create, including the mathematical concepts they use, is

also situated in the same space and time frame. What people create is usually referred to

as culture. Therefore, I could state that mathematical knowledge is culturally bounded.

12

Historically, mathematics as a cultural product was shaped by social events and
cultural differences. For instance, Bos (1984) pointed to those connections in a study of
how navigation during the sixteenth and seventeenth centuries and the rise of merchant
capitalism influenced the development of the Newton’s m In the same way, the
development of ballistics has to do with Galileo’s achievements on the parabola. In
short, social movements can provoke and shape developments in science in general and
in mathematics in particular (Kline, 1980; Machado, 1987).

Taken at large, culture can also be understood as a form of production. As pointed
out by Giroux (1988), "Culture signifies the particular ways in which a social group lives
out and makes sense of its 'given' circumstances and conditions of life" (Giroux, 1988, p.
193). However, one should note that these ways of life result from interchangeable
relations between specific social behaviors and structures of the society at large.
Therefore, they engender a historic-social process where culture can be both a product
and a process of production (Freire, 1985).

Bishop (1988) supports the idea of relating culture and mathematics education.
According to him mathematics is invented and created through social and interpersonal
interactions. It is dependent on cultural context, as well as being a cultural product. He
explains it by saying that there are six fundamental activities of the human being that are
necessary for the development of mathematical knowledge: counting, locating,
measuring, designing, playing and explaining. From these activities comes a symbolic
technology, one of the elements of culture which functions to relate people to their
environment and to other people. Other components of culture are ideological,
sentimental and sociological ones. Different symbolic systems were developed in
different cultures and different values. Western culture emphasizes the following values
in mathematics: objectivity, security, control, and mystery. When these values are
brought into the curriculum in mathematics education the emphasis falls on the idea of

“absolute truth," objectivity and formalism in mathematic teaching. Bishop argues that,

13

in mathematics education, we should move toward progress, rationalism and openness.
Students should construct their own meanings of mathematics content, rules of behavior,
goals, and so on. They should develop mathematical knowledge by sharing meanings
within the dynamic of classroom where the process of communication is fundamental.

D’Ambrozio (1985) relates culture and mathematical knowledge as he introduces
the concept of ethnomathematics:

Marking a bridge between anthropologists and historians of culture and
mathematicians is an important step towards recognizing that different modes of thought
may lead to different forms of mathematics; this is the field which we may call
ethnomathematics (D’Ambrosio, 1985, p. 44).

Ethnomathematics has been understood by some authors as the “study of
mathematical ideas of non-literate peoples” (Ascher and Ascher, 1986) or as
“mathematics incorporated in popular cultures (Ferreira and Imenes, 1986). In this sense
mathematics can be of a kind that is different from the academic pattern of knowledge
taught as mathematics. Ethnomathematics looks for “different mathematics” in daily life
of “different cultures” to reconsu'uct their practices into methods and theories (Gerdes,
1989).

From the studies in the field of ethnomathematics two basic contributions might be
highlighted. First is the attention given to the study of mathematical teaching and
learning in relationship to the whole of social and cultural life (Gerdes, 1988). Second is
the recognition of different modes of thought as having important implications for
approaching construction of mathematical knowledge. From these contributions, a third
proposition will be assumed.

W: Leaming implies consuocting knowledge.

Learning is a result of interactions between individual and objects within social and
cultural contexts. All three components, the subject, the objects, and their interaction, are

active elements in the process of knowledge construction.

14

Von Glasersfeld (1985) sees knowledge as a conceptual construction. Therefore
what is an object of knowledge should not be viewed as having a concrete existence, but
rather a conceptual viability within the experiential world of a knowing subject. Here,
the focus is on the activity of constructing. The knowledge that is created does not reveal
an independent world outside the mind of the knower. Implied in this more radical
consu'uctivism is that different theories and conjectures about the world are equally valid
until some criteria are established that allow for qualitative judgments and comparison
among conjecrures.

Lerman assumes that the rejection of certainty of knowledge, within constructivism,
does not deny the existence of a real world. Rather he recognizes that there is no way of
reaching absolute knowledge of the real word. Objectivity ultimately rests in the public
nature of language. Through language, meanings are negotiated in a particular culture,
time and place. What we know as theory is the result of our own attempts to organize our
experiential world (Lerman, 1989).

Addressing constructivism in mathematics education, Confrey (1987) also agrees
that knowledge is not “an accurate picture of the way things really are.” Because of that,
according to her, knowledge is always subject to modification. She assumes a ‘
constructivist point of view where students are not merely receiving knowledge but
constructing their own understanding of mathematical knowledge. The implication of
this epistemological position is to see the learner and his/her social interactions as central
elements in mathematics education.

W: Mathematical knowledge and children’s experience have a
unique encounter in the classroom setting.

This uniqueness could be described by taking into consideration the following
aspects. First of all, schooling is an event which marks a new phase in the child’s life. It
is organized, prepared to happen in such a way that expected outcomes will result from

the school experience. These results are not defined by or expected from the student. In

15

fact, he/she does nor have any participation in the decision making regarding what is
going on within the school. On the other hand, the event of being at school becomes part
of the child’s life, imposed by norms and patterns of the modern society where we live.
In addition, these school experiences might determine different directions, guidance and
influences in his/her whole life.

Second, in the process of schooling, the child is not a passive element. While a
student, he/she does not give up his/her condition of being a child. Within the classroom
setting, life outside of school cannot be eliminated. In addition, constructing knowledge
involves social construction where one of the elements is the individual him/herself.
Therefore, the culture of the classroom and the cultural processes embedded in the child’s
experiences contribute to the uniqueness of this encounter.

Finally, in the encounter of individual experiences of students and school,
mathematical knowledge in the classroom setting engenders not only idealized outcomes
but also conflicts and confrontations. The study of this encounter is of great relevance to
undersranding the construction of mathematical knowledge. It may unveil not only its

characteristics but also children’s strategies of coping with the encounter.

n n wl n n x
In defining the four propositions in my theoretical framework, I have organized the
literature I used into three groups: 1) studies with emphasis on student and teacher inside
the classroom context; 2) studies with emphasis on cultural and social differences in and
out of the classroom context and; 3) studies with emphasis on students’ strategies in

solving division problems independent of the classroom context.

Selected Studies of Classroom Learning in Mathematics

Several Studies can be included in the first group, varying from descriptions of

teachers' and students' beliefs to, more comprehensive analyses of classroom social

16

organizations and interactions. Next, I will summarize the bibliography in the first
group.

According to Ball (1988), teachers convey to students messages about their own
beliefs and understanding of mathematical knowledge. This means that teachers' beliefs
affect students. Since many teachers hold a view of mathematics as a body of skills to be
mastered through drill, the practice in mathematics class might frequently result in
teaching for mastery of procedures. Students learn by practicing and the teacher gives
the necessary structural support to this practice. This might result in some students who
are good at learning and following mathematics, algorithms and other students who do
not have the necessary capability to get to the best level of learning. On the other hand,
the view of mathematics as a way of abstractly working problems centered in real life
context might result in a very different style of teaching.

Ball (1989) discusses teaching for understanding as an approach that empowers
students to make sense of mathematical knowledge. According to Ball, students should
be able to reason with and about mathematical ideas. Teachers should be able to guide
this process through effective teaching based on knowledge of mathematics. Ball
disringuishes two kinds of mathematical knowledge: knowledge “of” and knowledge
“about” (see also Larnpert, 1988). Knowledge “of” mathematics implies knowledge of
mathematical concepts and procedures. Knowledge “about” embodies different
dimensions of the subject matter: its nature, how it changes the way truth is established,
pedagogical reasoning and subject matter in culture and society.

The latter view connects what students learn, and how they learn it, to the historical
development and the growth of knowledge as a constructive process of continual
invention and revision. Only “knowledge about” includes teachers’ understanding of
subject matter. In addition, teachers’ subject matter knowledge interacts with their
assumptions and beliefs about teaching, learning, students and context. These factors

shape the ways in which they teach. Thus, good teaching is grounded in the knowledge

17

about students, beliefs, subject matter and context.

Authors who hold this line of thought believe that students must be actively involved in
constructing their own understanding. Teachers help students to be independent learners
when their teaching allows students to invent, create, and judge the validity of their own
ideas and results. Lampert’s research is an important example of this philosophy of. In her
work with fifth grade teachers, she calls attention to the role that classroom culture plays in
the social construction of mathematical knowledge (Lampert, 1988). Her research was
conducted in a fifth grade classroom applying Lakatos' and Polya’s ideas of valuing conscious
guessing, inventiveness and risk taking. In her research, Lampert used reciprocal teaching to
establish a social climate within the classroom between teacher and students, as well among
students. The main goal was to establish a new model of social interaction in the classroom
where meanings could be shared and students could create mathematical arguments instead of
giving right answers. In addition, Lampert identified several patterns of the student’s
approach to learning situations including turning to the teacher as authority for verification
and treating rules and formulas as if they were arguments.

Peterson, Carpenter and Fennema (1988) investigated teachers’ knowledge of students’
knowledge in mathematical problem solving and found a significant relationship between
them. They found that more knowledgeable teachers use more active ways of teaching by
questioning students about problem- solving processes. Less knowledgeable teachers use a
more “passive way” of teaching by explaining to the students how to conduct the problem-
solving processes. While investigating teacher’s thinking about division Tirosh and Graeber
(1991) found that pre-service elementary teachers present “misbeliefs” about the division
process. The "misbeliefs" interfere with the way these teachers regard situation problems
involving division (Ball, 1990; Tirosh and Graeber, 1990).

Kaput’s work (1988) focused on students’ understanding of multiplicative
structures as ways of connecting concrete representations and mental processes. In other

words, he looked for mechanisms bridging the formal mathematical system and “lived in

18

situations.” He went from the students’ understanding of mathematical structures to the
construction of software “to facilitate the growth of this understanding” (Kaput, 1988).
Lanier and Anang (1982) conducted research in two ninth grade classes where the
social organization of the classroom and the subject matter knowledge were the foci.
They found that the relative position of the subject matter and the social organization in
the two classes affected the students’ opportunities to learn. How teachers organized
their interaction with students affected the students’ ability to ask questions, demonstrate
their abilities and engage in public performance. The amount of teacher-student
interaction also influenced students in learning basic skills, seeing the applicability of the
mathematical knowledge, and enjoying the subject and the class. Ball and Wilcox (1989)
also examined the interaction of context and content by comparing two in-service teacher
education programs. They analyzed teachers’ points of view about their assumptions of
what teachers should know and be able to do, and models of changing teachers’ practice.
The authors found that both programs were similar in their intent but different in their

context, curriculum, and their opportunities for teachers to learn.

Selected Studies of Social/Cultural Influences on Mathematics Learning

The second group of studies focuses on cultural and social influences on learning of
mathematics. Stigler and Baraner’s article (1987) is illustrative of that focus. They argue
that all cognition is highly contextual and domain specific. In this perspective, learning
in general, and mathematics learning in particular, is an inherently cultural process. For
the authors mathematics is a result of “culturally constructed symbolic representation
and procedures for manipulating these representation” Culture, tools, practice, and
institutions shape mathematical thinking and become part of the individual's repertoire of
mental representations. Stigler also demonstrates how characteristics of cultural belief
systems may help explain differences in achievement in mathematics of learning among

Japanese, Chinese, and American children.

l9

Lester (1989) studied the use of mathematics in everyday situations in contrast with
its use in mathematics classes. He found that in the everyday world people often use
mathematics procedures and thinking processes that are quite different from those learned
in schools. These differences include a focus on meaning, intentions, making sense, and
natural language in conuast to a focus on syntax (symbols and rules), formal language,
manipulation, and memorization.

Studies conducted in Brazil contributed to a belief that mathematical reasoning
outside of the school context unveils new sides of the thinking processes not revealed by
school mathematics (Carraher, Carraher and Schliemann, 1988). Based on Piaget’s ideas
they investigated everyday activities inside and outside of school looking at the implicit
knowledge in the way people organize their actions to solve problems, and at the role of
cultural contexts in this organization. The authors found that many students who fail in
school mathematics do very well in their everyday activities involving mathematics.
Children and teenagers construct complex models of mathematical reasoning when
working with lottery games, in markets, etc.... (see also Schliemann and Acioly, 1989;
Schliemann, 1990). According to Schliemann and Acioly (1989), “School experience
does not play any role in the ability to solve problems at work.” School experience
seems to affect more academic questions or problems are outside of work.

Ginsburg and Allardice take a different position (1984). They studied children’s
difficulties with mathematics in the social context of school. They distinguished two
systems of informal knowledge students use to solve problems. The first is tied to
biological components and appears before children enter school, while the second is tied
to cultural influences on students in pre-school years. Studying third and fourth grade
students, the authors concluded that students have great ability to use informal knowledge
and invented procedures. However, the students produced consistent errors, arbitrary

results and misunderstanding of mathematical principles. The authors suggest that the

20

basic questions on which researchers should focus involve the cognitive processes of
informal knowledge, invented procedures, and errors.

Onslow (1991) attributed students’ difficulties with school mathematics to their
lack of ability to construct links between the abstraction of school mathematics and its
real world context. Students are knowledgeable in mathematics in everyday situations,
but fail to understand the same basic principles when presented in the form of symbols.

Rogoff and Lave (1984) bring a special conuibution to the discussion of the aspects
of everyday cognition and its development in social context. They argue that thinking
and context are interrelated and, thus, social context affects cognitive activity. Rogoff
and Lave define context as physical and conceptual structures as well as the purpose of
the activity and the social milieu in which it is embedded. According to Rogoff and
Lave: “The person’s interpretation of the context in any particular activity may be
important in facilitating or blocking the application of skills developed in one context to a
new one” (Rogoff and Lave, 1984, p. 3).

Studying the activity of grocery shopping, Rogoff and Lave (1984) looked at the
nature of arithmetic activity related to problem-solving situations in the supermarket.
They described several characteristics of arithmetic problem-solving in the supermarket
in which procedures of solution were dialectically constituted. The operations used
created circumstances of continual monitoring of results which might explain the virtual,
error-free, arithmetic performance of shoppers in contrast with their results on formal
testing situations.

The third group of studies on mathematics' teaching and learning is related to the
students’ strategies to solve division problems. The bibliography in this third group will

be highlighted in the following section.

21

Selected Studies on Students Doing Division

The sequence of studies I will present next was chosen based upon two criteria.
First, my research was conducted in a third grade class at an elementary school. During
the time I observed mathematics classes, division was the unit being studied. Thus, I
selected studies approaching mathematical teaching and learning in general and division
in particular. Second, I was interested in learning more of what the literature had to say
about students‘ strategies to solve division problems. Thus, I selected literature on
research dealing with everyday cognition, focusing on a student’s own way of dealing
with those situations.

The study of the student’s strategy for solving problems is not new. In 1981, a
conference was held at Indiana University entitled "Issues and Directions in Mathematics
Problem Solving Research." Several papers were presented and lines of research for
mathematics education were discussed. Mayer (1981) argued the importance of
establishing basic principles of learning and cognition to promote meaningful learning. He
distinguished algorithmic procedures from representations used by the students in problem
solving. The author suggested that problem solving involves a series of mental operations
that transform knowledge representation rather than a series of learned behaviors.

In a paper about priorities for mathematics education research, Lesh and Akerstrom
raise an important issue on applied problem solving. The issue is the adequacy of word
problems when related to everyday situations that students deal with in the real world.
The authors claim that word problems often differ from real world problems with respect
to the degree of difficulty, the processes needed in the solutions, and errors most
frequently made. These and other presentations reinforced the relevance of the theme of
solving problems in future developments on mathematics education. The last decade has
been particularly important in the development of studies involving multiplication and
division in solving word problems. There is a general agreement on the constructivist

approach to the knowledge acquisition, that underlies these studies.

22

The importance of the active role of the student in constructing mathematical
knowledge is taken for granted. Two lines of development seem to emerge from that: 1)
the recognition of the importance of the cultural and the cognitive aspects on learning
process and 2) the relevance of the student’s informal knowledge about mathematics.
Common to both is the recent effort researchers have made to look for insrances of
mathematical reasoning outside the school setting. In addition, the relationship between
mathematics taught at school and the mathematics students deal with in real world
situations has become the focus of attention.

While investigating the levels of mathematical understanding shown by British
secondary school students, Booth (1981) reports that children operate within a system of
their own. They do not use the mathematics taught at school but, rather, the mathematics
belonging to a universe different from that of formal mathematics.

There is no complete agreement about the origins of a personal system of operating.
Borel (1987) uses the term "natural logic" to contrast the child’s logic with forms that
logicians know. The former is characterized as being real logic and natural thinking or
naive logic and naive thinking. The latter is characterized as being formal abstraction,
pure axiomatic or refined thinking, all of which are very specialized constructions of
logicians.

In this regard, Fischbein’s studies have been largely quoted in the discussion of
ways children solve division word problem. Fischbein argues that children’s work is
mediated by implicit, unconscious, and intuitive models. He defines intuition as part of
intelligent behavior with characteristics of immediacy, globalism and extrapolative
capacity, among others. He further differentiates primary intuitions derived from the
children’s own experience without any formal instruction from secondary intuitions
formed mainly at school. He believes that these intuitions of both types, being natural
and basic or reflecting the way it was taught in schools, are resistant to change and

intervene in a child’s attempt to solve a problem. In one of his many publications,

23

Fischbein (1985) defines intuitive models associated with the four arithmetic operations.
For division he distinguishes two models according to the structure of the problem. First,
in partitive division, or a sharing-type of division problem, an object, or collection of
objects is divided into equal segments or sub—collections. Second, the quotative division,
or measurement, is defined as how many times a given quantity is contained in a larger
quantity. According to Fischbein (1985), the model imposes the following four
constraints on the search for a solution to the problem.

1. The dividend must be larger than the divisor,

2. The divisor must be a whole number,

3. The dividend must be larger than the operator,

4. The result must be smaller than the dividend.

While studying the invariance of the operation modeling a situation and the
numbers involved in division and multiplication Greer (1988) examined Fishbein's
arguments, he questioned why these intuitive models continue to affect students’ thinking
in more general domains. The author suggested that both the student's inadequate
experiences and lack of attention to mathematical modes of thinking contributed to that.
He also suggested that “hard, ” or more difficult, numbers should be used routinely in
problems, as well as creating cognitive conflicts and reducing single Operation word
problems.

Kouba (1989) and Bell et al. (1989) illustrated some of the limitations of
Fischbein’s theory of implicit intuitive models and present some contributions to this
discussion. Kouba examined the intuitive models to derive a classification system for
children’s suategies by testing models of development of solution strategies for
multiplication and division problems. She worked with more than one hundred children
fiom the first, second and third grades and she concluded that “a more detailed definition
of children’s behavioral interpretation of division” was necessary to clarify Fischbein’s

intuitive models. Bell et al. studied more than 300 subjects ranging from 10 to 20 years

24

in age to review Fischbein’s theory on multiplication problems and propose a new one.
The authors claimed that there is insufficient consideration to pupils’ numerical, rather
than structural, perceptions in Fischbein’s work. They developed the notion of
“preferred multipliers” and they demonstrated how children and adolescents are sensible
to the structural aspects of multiplicative problems by their success in estimating
answers. Therefore, the numerical preferences and other factors dominate their choice of
the operation to be used in a given situation. This study suggested that the same
questions should be asked relating to division problems. In 1984, Bell et al. published a
study about the effect of the structure of the problem and other factors on the choice of
operation in verbal arithmetic problems. They worked with children 12 and 13 years old
and observed that the children reasoned in a qualitative way about problems, but were
unable to relate the numbers to the problem quantitatively. The students did not
perceive, for a given context, the invariance of the relationship among the numbers
despite their size. In addition, uncertainties about language and notation were identified.

Bell and his colleagues found that the type of numbers in the problem was a source
of difficulty, especially when the operation of division or multiplication involved
numbers less than one. They asserted that a hierarchy emerged related to the struture of
the problem. Children moved from partition to fractional partition, to quotation, to
fiactional quotation and finally to rate. The authors found that the students made
successive attempts to transform the problem into more accessible structures to be
solved.

The influence of the context was also studied by O’Brien and Cabral (1989). They
were interested in lookin g at the achievement of students in first, second and third grades
on multiplication and division problems, in two different environments. Eighty-nine
students were interviewed to solve problems mentally and using paper and pencil. They
found that the environment was not a factor in determining achievement. Environment

in this study was defined by the environmental condition within which students had to

25

solve problems: with or without paper and pencil. However, there were significant
differences in performance, according to the structure of the problem and the students'
grades.

Teule-Sensacq and Uinrich’s study ( 1982) also focused on students’ achievement.
They suggested that working division problems within a situation where the students
might face the new could result in more success and variation on solution procedures.
This methodological procedure was described in terms of promou'ng dialectical
discussions of student’s actions facing the new.

Besides discussions about intuitive models, the interest of many researchers has
been the variety of strategies students use to solve division problems. These strategies
are rich in students’ mathematical reasoning to solve problems which do not necessarily
come out in the school context.

Generally speaking, it has been recognized that young children are good problem
solvers (Moser and Carpenter, 1982; Gilbert and Leitz, 1982; Carraher et al. 1987; and
others). Children facing a situation of problem solving come up with a variety of
strategies by themselves and are persistent and creative in working out solutions. They
make rational choices among strategies to solve particular problems and try to bring them
to a more concrete situation, more directly related to their own experience. Research has
been done to identify which factors influence their choices of strategies and to have a
better understanding of the children’s constructions (Zweng, 1964; Sowder, 1988 and
1989; Kalin, 1983; Weiland, 1985; Kouba, 1989; Boero, 1989; Schliemann, 1990; and
others as reported before).

Sowder (1988) has a different view of children’s strategies in solving problems.
Studying about 70 students in middle school, he concludes that students “might be
solving problems using strategies which are quite limited in applicability.” They do so
because of the emphasis on a computation-centered curriculum where the correct answer

justifies the means. The author asserts that a correct answer is not a “safe indicator of

26

good thinking.” According to Sowder (1989), children’s solutions to story problems are
immature strategies. He names three types of strategies: coping strategies, computation-
driven suategies and strategies of limited usefulness. There is no complete agreement
about the origins of a personal system of operating. He indicates that some of those
strategies are guessing the operation to use, trying all the operations and choosing the
most reasonable answer, and looking for isolated key words or phrases.

On the other hand, several other researchers have found very rich procedures
among the strategies children use to solve problems and sometimes complex mental
calculations, too. Among the more frequently named are: strategies they found sharing,
counting, repeated additions and subtractions and recalled facts. In 1964, Zweng had
already identified sharing and counting strategies among second grade students who were
solving division problems. While working with second grade students using a set of
blocks, Keranto (1984) found counting and sharing strategies. In partition division,
children used trial and error procedures to try out the size of sub- groups, or give out one
at a time to form sub- groups. More recently, among others, Kouba (1989) and Boero et
al. (1989) described, the use of the same manipulative distribution strategy based on
concrete material or drawings. Kouba discussed “double counting” and “transitional
counting.” In the first situation, students dealt with two counting sequences. They kept a
running count of the total number of objects in the groups. At the same time, they kept
counting out the objects to form the group. In the second situation, students counted
sequence based on multiples of a factor in the problem. Kouba also made reference to
the fact that among students from the first, second and third grades, the majority in his
study made use of a uial-and-error strategy. They guessed or estimated a number and
checked it. If it didn’t work, they uied another number.

Boero and colleagues, working with six-to 11-year-old children also mentioned trial
and error as an approach to use with successive addition and subtraction. Repeated

addition and subtraction, as well as recalling of facts among others, are always present in

27

the strategies children use to solve division problems (Boero, 1989).

Kalin (1983) used the expression “multiplicative thinking strategy,” referring to
recalled number facts by taking the multiplication as the inverse of division. Thus, 16
divided by two is equal to eight because eight times two is equal to 16. The author also
adopted the term “solution strategy,” when a student takes the known fact to find out the
unknown fact. Boero et a1. (1989) described this strategy as being part of the distinct
trial-and-error strategies, in partitive division problems. Kouba (1989) described recalled
number facts as the maximum grade of abstract procedures children take to solve
multiplication and division problems. The author argued that, in this strategy, childrens
do not use physical materials. They get the answer by recalling the appropriate
multiplication or division combination or derived fact.

Besides the strategies mentioned above, some authors refer to other suategies children
use when solving division problems. Weiland (1985) describes children using disuibutive
algorithm as shown in the following situation: 927 pennies to share with nine people.

900 pennies - 100 to each

27 pennies - 3 to each

Answer: 103 pennies each

or
927+9=(900-:—9)+(27 +9)
= 100 + 3
= 103
Finally, Carraher et al. (1987) studied written and oral mathematics to investigate
the effect of the situation on the choice of procedures and efficiency in problem solving.
The students were from third grade solved problems in three different situations:

simulated store problems, word problems, and computation exercises. The authors found

28

that there was no uniform strategy for solving problems among the situations studied.
Furthermore, the children had a good understanding of the decimal system, but the
educational system led them to focus on algorithm calculations. In oral mathematics,
they found two types of repeated grouping and decomposing. Children usually preferred
to deal with hundreds, tens and units in the opposite direction to that used in algorithm.
Also, they tended to work mentally with quantities ending in one or more zeros because
they were easier to work with. During the computational procedures children kept
monitoring the quantities with which they dealing.

Such diversity of strategies used by children solving problems raises the question
about how students make sense of mathematics. In the next chapter, I will discuss some

cultural and social components in children’s sense making.

CHAPTER III

CHILDREN’S PERSPECTIVES

Illumination

How much and in what ways can cultural background and social environment shape
students’ understanding of mathematical concepts? This is one among many questions
that could be posed when the subject is the social construction of mathematical
knowledge. To get to the answer, I believe it is necessary to know more deeply how
social and cultural forces affect learning of mathematical concepts. That is, how may
cultural background and social influences help explain the students’ construction of
mathematical concepts?

In this research, my main purpose was to unveil some aspects of this encounter. To
do this, I worked both inside and outside the school buildings. First, I tried to capture
moments in which students dealt with mathematical situations where social connections
could be made spontaneously by the students, or provoked in some way. This happened
in a variety of situations within the school, in mathematics classes, in the journals we
exchanged, in formal interviews, and in conversations at lunch time or other school
activities. This happened also outside the school, in the students’ houses, on the streets
during our conversation, at play, and at the stores. I uied to keep my mind open to
detect moments of the students’ mathematical reasoning in which their social or cultural
background made a difference in how students dealt with mathematical knowledge.

Second, I uied to create opportunities in which I could collect data about the
students’ perspectives on social themes and the mathematics they could relate to these
themes. I used pictures from local newspapers showing scenes of the students' social
context. Initially, I asked them to describe those scenes and comment on the picture.

Then I asked each one of them what he or she could see, create or invent about

29

3O

mathematics, related to the pictures.

Ten pictures were selected from those which showed scenes taken from the
students' milieu. The selection was also based on the idea that the picture should
challenge a student’s mathematical thinking. All of the pictures should portray social
events common in the students’ daily life. The themes in the picture were not defined
beforehand, but they were brought up during the interviews. The following are the
scenes in the pictures used in the interviews:

1- A car passing a sign indicating 60km/h speed limit

2- Several men from a paving company repairing potholes in the street

3- Several musical instruments and pictures of two men talking

4- A saint’s image being carried by four men and a crowd gathering in the church

surrounding the image ‘

5- A parking lot behind a building under construction where several men are

working

6- An ex-president, running for senator of one of the states located in the north

region of Brazil, leaving his campaign committee

7- A water reservoir with three pipes in it

8- A building with the word “Greve” (strike) painted on the glass door

9- A candidate running for governor of the state of M inas Gerais being interviewed

10- Brazil’s current President in a meeting at the Presidential Palace

The students in my Study were third graders in an urban school with 1100 students.
They lived in a community of unskilled workers in a medium-sized town located in south
east Brazil. The students ranged from nine to 14 years of age. Five of them were
considered out of the expected range of age for this grade. The number of family
members in those students’ household averaged six people. The school was located 10 to

20 minutes away from their houses in the majority of the cases. There is no public

31

transportation specially provided for the children. They go to school on their own and by

foot.

'1 ’ 'v A Th ir i l Milli

To give the reader a better understanding of the students' perspective on the milieu
where they live, I will report here how they expressed themselves in the interviews, as
they reacted to the pictures. As I said before, the pictures were used to stimulate
students’ thoughts and expression of themes related with the social and cultural context
where they live.

When I asked what the picture was about, the students gave a detailed description
of what they could see. They also added some kind of action or evaluation of it. For
example, if students were talking about the car in picture #1, they would say that it was
running, or the road was full of holes or dirty. It was through these kinds of comments
that I identified important social themes as part of the students' daily experience and their
way of expressing themselves.

Several themes were identified in the interviews including: religion, death,
protection, politics, inflation, power, authority, salary, food, and work. I will describe
them next in three major categories. I chose the categories based on the centrality of the
theme in relation to others, as they were discussed by the children. For instance, poverty
came up when students talked about the power of deciding about things in picture number
10 (the president of Brazil). Then “poverty” was subsumed under the theme “power.”

The three major categories are: religion, security and power. Under the category
of religion are the themes of death and protection. Security covers mainly the theme of
violence. Power embraces several themes discussed by the children in the interviews,
including: poverty, inflation, workers, jobs, salaries, strikes, shelter and elections.

Religion. Religion is one of the themes mentioned before. Traditionally, the

Catholic church is predominant in this region. According to this tradition, there is

32

emphasis on the initiation of children in the Church's rituals such as baptism, first
communion and others. At the age of seven or so, children go to the church school to be
prepared to receive first communion. That is a special day in their lives. However, there
is no emphasis afterward on continuing to perform this ritual, or even going to the
church on Sundays as their parents usually do. Talking about picture #4, they expressed
the idea of religion mixed with feelings of gratitude, protection, hope, sadness, and death.
The strongest connection they made was with death. They mentioned it several times.
The following are some examples.

Carlos: There is a coffin, a saint all illuminated... They might be praying for the

dead....

Laura:... praying for a friend of them because he used to do many good things to

them.

Vera: I see a “velorio” (deathwatch), a saint, and a lot of people praying. They

are praying because a relative is dead... to get salvation... it is sad....

Death was also a subject approached by the students in our conversation through
journals. I asked them to tell me about some experience they had outside of school. They
reported very traumatizing personal experiences, several of them involving the death of
someone. The following journal enuy is an example:

I remember it as if it was today. I was four years old when in a rainy day this
happened. It was raining, and my brother who worked in the country club, he
loved to work there, but that day was really raining and my mother did not
want to wake him up to go to work. When he got up on his own he was mad
at my mether because she did not wake him up to work. She said to him:
'Jorge don’t go there today it is too rainy.‘ He answered: 'Mummy I like to
work there, and besides when it rains we just keep chatting while it rains and
go home at the end of the time with our money.’ My mother insisted but he
wanted to go anyway. By the lunch time, my little brother was in his cradle
and my mother was fixing the lunch. My other brother came and asked for J.
and just while my mother was answering that he was about to come home
from his work my cousin rushed into our house. Something horrible

happened, she said. And then she told my mother that my brother fell down
from the pinguela* when he was going to work. My mother was crying

33

desperately. They called the firemen and the police but they did not find the
body. Even after the rain stopped and the waters lowered they did not find
my brother. They never have found anything.

This kind of “drama” is part of their daily lives due to the poor conditions in which
most of them live. Children and their families look to the Church for support. Because
of that the Church assumes two roles: 1) a buffer between the people and the struggles
they face, and 2) dispenser of protection and improvement for their lives. It was
apparent to me that when children talked of the Church they were not referring to the
institution, but to the power of changing things by praying to the saints. Rituals such as
mass and processions are performed to mediate their hopes and requests to the saints.

This is indicated in the following piece extracted from one of the interviews:

Eduardo:... People are attending a mass... there is a saint and they are praying...

Neuza: I think they are praying to Nossa Senhora (Our Lady). I think they are

praying to be protected... to improve life.

Silvio: Probably they are asking (Our Lady) to provide for food in their homes... and

not let it run short for others, too....

In general, there is a promise made to the saint by those asking for grace whatever it
might be. It means that a promise is made to the saint as way of “symbolic payment” or
thanksgiving for things they receive. The following conversation illustrates this point.

Ruthz... a woman is married and her husband drinks a lot, so she goes there (to the

saint) and asks him to make her husband stop drinldng and then, he stops. When he

stops, she... my mom has already done the promise... she took white lace to the saint.

Leda: Every year I go to Aparccida do Norte (a place where the saint is who is con-

sidered a national protector of Brazilian people). I, my grandmother, and my aunt...

 

* A pinguela is a connection between two hills made with a piece of wood, usualy a cut tree. It is used
most often to cross small rivers or flooded places during heavy uopical rains. It is not normally
dangerous. However when it is wet, a pinguela is a deadly trap.

34

this year all my family is going to....

Interv: Do you know why your mother made the promise?

Ruth: I know. It is because she was sick and had to have medicine in the hospital. She
asked the saint to cure her. When she got better she went there and gave her (the saint) white
lace.

Snoozing. In the picture #6 there are several cars in a parking lot, including a police
car. This was enough to initiate a conversation about “security.” Children emphasized
the role of the policeman in controlling crime and protecting people against robbery,
assault, and other criminal acts. Children expressed themselves as follows:

Lucia: There is a police car, other cars and a wheel barrow... I think that there are

men working there taking soil out of the excavation.

Interv: Why is there a police car?

William: The policeman is there to see if anyone exceeds the limit allowed by the

traffic signal... to obey....

Fernanda: Do you know what I think about this? The police might be hunting

some criminals or kidnappers...

LE: There is a policeman there... might have been an assault...

In the large cities in Brazil, violence and crime are major problems and a great
number of the population live in subhuman conditions without adequate food, clothing,
and shelter. The city where this research was done is the third largest city in the state and
is not far from two big capitals, only 300 km away. This proximity is one factor in
facilitating the increase of crime in the region, because it has been a place for robbers and
other criminals to hide. While I was collecting data, three events gained the attention of
the whole p0pulation, one of them having national impact. All three happened in the
school neighborhood. The first one was the kidnapping of a police officer by three
prisoners. They kept the policeman in a house for days until the kidnappers surrendered.

This caused change in the routine life of the whole city including the school. All school

35

activities were suspended for three days to preserve the children’s security. The second
event was the murder of a man, and the third, a rape of a fourth grader student fiom the
school where I was collecting data. I heard from neighbors that this was the third rape
that had occurred that year to a student from this school.

I should say that there is apolice station adjacent to the school’s main enuance. Every day,
studentspassbyitastheyenmrthe school. Thisstationisopendayandrrightbuthasfew
personnel. This limits theamountofassistancetheycangivetotheneighbmhood. Many times
the police are called to deal with cases of alcoholism involving men beating women and other
violence. The police deal with ongoing family violence, but their action is peripheral to
enhancing the actual security of children and families, and community security is by no means
guaranwed.

flower. These passages have shown how some specific pictures gave rise to various
themes including religion, prorection, security and others. I would like to stress that
among the 10 pictures, seven brought up themes related mainly to politics, power,
inflation, poverty, work, and, salary. In addition, I would like to address the relationships
of students' comments to these and other themes. The following Figure illustrates how I

see the power relationships students perceive in the society as a whole.

SAINTS

P“ S I DENT
GOVERNR

PRINC IPLL
POLICE

 

JOB I'OOD PROTECTIOI’
normals SALARY

mm EDUCATIOI
9910:1131 poopl.
give to not t or to the
people povertull

 

Figure 2 Power relationship students perceive in the society

36

Children associate the power to do things, give, decide, divide, take and so on with
someone else above them such as the mayor, the president, the governor, a policeman,
and saints. More than that, the power is not related to institutions, for example, the
church or the government per se. Power is personalized in the figure that represents the
institution at that moment. Power is not perceived as residing in the government. Instead,
it resides in the official such as the governor. It is not the church as an institution but the
saint who has the power. The saint, the governor, and the mayor have the authority and
power to decide and solve problems. On the other side are those who do not have the
power to decide, to have or to do anything, and who as a consequence, ask for and wait
for those in power to act. The powerless are people in general like themselves, the
workers, the poor. This condition of inequality is expressed often by the children when
they talk about workers, salaries, jobs, and housing and other issues. The following are
some examples:

Eduardo was talking about what he thought was not good in picture #2.

Interv: Eduardo:... people working for nothing.

Do you know someone who works without payment?

Eduardo: No, working for norhing... I am saying... doing a lot of effort and

getting almost nothing.

Interv: And here (in the picture) who do you think has low earnings?

Eduardo: The workers (showing the men who were working throwing asphalt).

In another moment of the conversation in picture #4:

Eduardo: In this picture there are several people and there is a saint... they went to

visit the saint.

Interv: Why did they go to visit the saint?

Eduardo: Because the saints are the protectors of the poor.

Nclio: Here there are several people. There is one... kind of window with some

flowers, and there is a saint. I think they are praying to Nossa Senhora (Our Lady)

37

Aparecida. I think they are praying... to protect them and to improve their lives.
Rosa: They might be asking nm to fall short of food in their homes... for other
people not to fall short of food either.

In picture #5, about the former president:

Hilda:... he could help the poor, and not increase the prices too much.

In picture #8, about the former state governor:

Fatima: He is being interviewed by the press. To be a good governor he could help

poor people and abandoned children.

Interv: If you were the governor what would you do?

William: I would give more food, more houses, and schools.

Laura: It is a bank on strike...

Inter: Why are they on strike?

Roberto: Because B. (the mayor) doesn't want to give them money.

Lucia: To get more money... to have more money....

Interv: Money for whom?

Lucia: Ah! For us! For everybody who works.

Interv: Why do you think the mayor doesn’t want to give the money?

Roberto: Because he is selfish.

One important aspect of this picture is that it was the only one where children
mentioned some kind of social movement to get something from the established power.
The word strike was in the picture and they related it to several strike movements that the
community had seen lately: workers in banks, teachers from public and private schools,
and students and teachers from the University. In our conversations about strikes none of
the students mentioned or referred to a suike as a movement of workers asserting rights.
Some students mentioned dissatisfaction of workers with low salaries, like the teachers’
salaries, or not having wage increases equal to the inflation rate. This might suggest that

the students see a suike not as a movement for requesting something that is their right,

38

such as fair salar. Rather, the students see it as being a response, or a reaction to the
unfair condition of low salary. In this sense, the worker stops working because he or she
does not get a good salary.

Children expressed their understanding of a suike also as a result of dissatisfaction.
Workers suike because they are dissatisfied with their salaries. As mentioned earlier, the
children gave several reasons to justify the strike but the workers feeling inferior to their
bosses was also present in their comments. Once more I noted the personification of the
figure of power. Here are some examples:

Luiz: The bank is on strike. There are some people working and others outside.

Interv: Why was the bank on suike?

Luiz: Because they were tired.

Fabio: They were on strike because people in Brasilia (where the central govem-

ment is) do not send money to pay people who work here... and then they go on

strike. And then while they don’t have this increase in their salary, and the same
for the bank workers, they don’t go to work.

Williamz... strike is not to work for some days...

Interv: Why?

Creuza: because the Mayor is paying low salaries...

Williamz... they are angry with their boss... because they pay low salaries, the

same with teachers.

Interv: Why does a person go on suike?

Fatima: because the president doesn't give them better salaries.

Among the themes more often addressed by students during these discussions were
low salary, poverty, inflation, housing and food. Picture #10, where the Brazilian
President appeared to be talking to someone, provided a particularly rich oppportunity
for children to express their anxiety about these themes. Children showed that they
believe in the person who is the president. They believe that, by being a president, he

39

could solve problems. The following conversation about important things students said
the president should do exemplifies this and reinforces the point about power and
inequality I discussed before.

Rosa:... he should build a village with a lot of houses and give them to the home-

less to live. To give food to them....

Robertoz... ah Maria, to give them a job too.

Laura... one who cannot buy anything, does not have anything to eat, too...

money... because it is misery what they are paying... It has to improve... the

government, Brazil. For example... can you imagine everybody... even doctors,
electricians... everybody could have one hundred thousand per month?

Roberto: (surprised!) GOSH!

Laura: Ah, that is for everybody to be rich like them... Isn’t he rich?

Do you think he earns 100 thousand?

Laura: Ah! more than that...

Interv: How much more?

(Everybody answered together in chorus): MORE than one billion!!!

Nobody talked about elections as a process of choosing their representatives,
although they were in the middle of a political campaign. Within a month from the time
I collected these data, elections would be held for the governor, congressman, and
district representatives. In addition to that, there was a presidential election a few
months earlier after 20 years of dictatorship. Both events seemed not to have inu'oduced
anything new to the children’s lives. In fact, lately in this country, politics were seen as

an opportunity for corruption.

To learn about the social and cultural factors in the construction of mathematical

knowledge, I examined two dimensions: the students’ social context and their

4O

perceptions of school mathematical knowledge. This examination was necessary for two
reasons: 1) to be able to identify them when together, and 2) to perceive their differences
in the construction of mathematical concepts.

Having examined the students’ context, I will now present some general findings
about how students perceive mathematics. After that I will analyze the contrasts I see
between occasions when students talk about mathematics and when students do
mathematics in their daily activities.

First of all, I have to distinguish here two different aspects of the data: when
students are “talking” about mathematics and when they are “doing” mathematics.
These moments do not represent a sequence in the students’ responses, and they are not
identifiable as hierarchical levels. These moments are only different instances in student
responses or behavior. To make this clear, “talking about” mathematics means students
referring to mathematics as a third element in the conversation. They are not talking
about themselves or about something related to them. They talk about something outside
of themselves. “Doing mathematics” means students are reasoning in mathematical
terms or analyzing any kind of mathematical situation without thinking about whether
their activity is mathematics. Talking about mathematics typically happened when
students and I were talking about mathematics classes, or in their journals, or talking
about mathematics in the pictures I used in the interview, or about mathematics in their
games and other activities. Doing mathematics, on the other hand, happened when
students were doing things requiring mathematical reasoning, inside or outside school,
such as organizing teams to play queimada, scoring the game set, solving school
problems, cooking, or helping their parents.

Initially I address mathematics as students talked to me. Later in this report I will
be referring to “doing mathematics” to conu'ast with the first one. On different occasions,
we discussed this subject: in class, in their journals, during the interviews or in our

conversation during their games, on the street, and so on.

41

Here are some of the definitions they gave me when I asked them, through their journals,
what mathematics meant to them:

Mathematics is...

number, calculations...

calculator, computer, television...

the number of things.

things we learn... the numbers...

calculation and problems we solve... divisions or things like that.

number of persons, desk, shoes... written numbers also, ordinal numbers, cardi-

nals, fractional, multiplication, etc...

a subject matter to learn how to work as fare man, driver, teacher...

a subjecr matter we learn at school which has calculations, division, multiplica-

tion, etc...

The message from these definitions is clear. According to the students’ responses,
mathematics is nothing but numbers and calculations. The meanings expressed here do
not go beyond counting, calculations, and some elementary and meaningless operations.
I say “meaningless” because students only named them without any context that could
say more than numbers.

The idea of mathematics as being the expression of numbers became quite strong
during the interviews. Using the pictures described earlier, showing scenes of the
students’ daily experience, I asked them to tell me what they could see, create or invent
about mathematics related to those pictures. The result was astonishing: The majority of
students pointed out to me almost exclusively the number of the picture. I have to
explain that each picture was over a white sheet of paper showing numbers from one to
10 to identify each one of them. Many times, when I insisted that they take a careful
look at the scene and tell me something more, they found a smaller number on the top of

the picture, the date when it was taken. Sometimes I got answers such as the following:

42

Eduardo: There are several u‘ecs... two pipes, adding this one, are three...

Rita: One car adding another car is equal to two cars...

Sonia: There are six doors... they are divided in four parts... There is a man sitting

on a chair with four legs...

Beto: The number 60... (in the picture showing the speed limit).

Nelio: There are seven men and the number seven is odd.

Edmar: There is the number of the license plate on the truck. Within the truck

there is a watch, the thing that shows the velocity of the truck and has numbers in

it.

Ana: The truck has four tires and the number four is mathematics.

Hilda: There is money they can put together, count, do things...

According to these students mathematics was numbers shown in the pictures and in the
control panel of cars, numbers of people or things in the picture and, more rarely, some
calculations they created based on the number of objects in the picture.

I found the same pattern of response when students and I were talking about
mathematics in their activities outside of school. In this situation, the most common
answer was to mention the time when they were doing homework at home or when they
were playing school with their friends. Besides that, students were not able to identify
mathematics related to the pictures or in their activities outside school or at home.
Exceptions could be made in the activity involving money, although students did not
mention that in the discussion about the pictures. In the conversations we had, students
made several references to the activity of dealing with money at the stores, in the bus or
in others situations where they had to pay or receive money. Those situations were
quoted as examples of opportunities where students were dealing with mathematics
because they have to perform calculations and were dealing with numbers.

The following dialogue about mathematics between two students and the

interviewer shows how difficult it was for students to relate mathematics to things other

43

than money and numbers. Part of this dialogue deals with a woodworker who is the
father of one of the students and occasionally is assisted at his work by the student.
Cid: Mathematics helps you not to get in trouble. For example, if you are going to
work as a bancario (teller in a bank), you’re gonna divide money to give people
or... as trocador (bus fare collector).
Betoz... or as a teacher.
Interv:... how is that?
Beto: IfI were a teacher, how do I know how to divide to teach the students?
Interv: Does your father know mathematics?
Eduardo: Yes.
Inter“. Does mathematics help him in his job?
Eduardo: No, he is a woodworker. He makes furniture.
Interv:... and mathematics does not help him to deal with his work?
Eduardo:... dividing nails, for example... There is a drawer full of nails. When the
nails drop on the floor... if the drawer drops on the floor, he gonna know how to
divide... no, not to divide like that... no, no, it doesn’t help anything.
Interv: That is interesting. This kind of mathematics that helps on the job but
doesn’t help your father....
Eduardo: Well..., it helps only if it is the trocador (fare collector) or the banker,
and the teacher.
Beto: The bancario? (teller in a bank).
Eduardo: Yes, the bancario!
Interv: Ah, it is not the banker?
Eduardo: Banker is the owner of the bank and bancario is the one who works in
the bank.
Interv: So, there is no mathematics in the wood workshop?

Eduardo: None. Only when his work is paid for, or when he is going to make

change.

Mathematics was identified with numbers by students also in their journals. I had
explained in the beginning of my contact with them, that journals would be an insuument
for us to establish a conversation about mathematics. I told them that I would like to have a
conversation about their experiences with numbers in their games, at home, or at school. I
asked them to show me with drawing how they saw mathematics in their games and other
activities. Many of the students started copying and making calculations involving basic
facts of division, multiplication, subtraction and addition. They copied exercises from
books they had at home. Sometimes they just copied the table of results of the four
fundamental operations. Other students expressed themselves through drawings like the

following:

 

 

Figure 3 Mathematics class

45

The student drew a classroom situation where the teacher is explaining how
students should divide numbers. The teacher is saying: “I will explain to you how to
make this calculation because 9 times 3 is 27.” In fact, the student was representing
mathematics as the subject knowledge the teacher teaches at school. The real situation of
mathematics class represented here shows the teacher relating division with
multiplication. Note that the student repeated teacher’s words exactly but not the correct
symbolic representation. I would speculate that the student was repeating the teacher’s
words but not reasoning in terms of the mathematical operations involved.

Despite this evidence, some discrepant cases were identified. The following cases
show students talking about mathematics as operations, not just as numbers or money.

The first case was a student who began her journal with the picture in figure 4:

_-

  
   

(El

 

 

 

 

Figure 4 Balloons

46

She explained what the picture was about:

I see mathematics in the price of the balloons and in subtraction because the

woman who is selling them had nine and she gave one to the little girl and kept

eight.

In an informal conversation, Fatima and her br0ther told me about things they do
to help their mother at home which involve mathematics. Part of the dialogue is as
follows:

Fatima: I do the dishes and carry water fiom outside.

Inter: How much water do you bring each time?

Fatima: I bring two full gallons.

Interv: What mathematics can you see in carrying water?

Fatima: Carrying two and two, four.

Dair: I know. It is when one can is heavier than the other one.

In a variety of our interactions the students identified mathematics with numbers
rather than the meanings they represented. As one can see, those children did not
perceive mathematics as part of their everyday activities. Students did not relate
mathematical knowledge they study at school with mathematical knowledge they deal
with outside school.

Studying the students' perspectives on mathematics and social themes related to
it, I learned that the context where students do mathematics is crucial to understand
the kinds of mathematics the students use.

In the next chapter I will approach the contexts where the students face
situations involving mathematics in and out of school. These were cases where
students were reasoning in mathematical terms about situations happening in their
daily life. However, for the most part, there was no evidence at all that children see

school mathematics as related to their lives outside the school.

CHAPTERIV

CONTEXT S OF CHILDREN’S EXPERIENCES
INVOLVING MATHEMATICS

Innodusztian

I conducted my research in a urban public, elementary school in the same city of
almost one million inhabitants, in southeast Brazil, where I work at the University. The
school is supported by the city’s Secretary of Education, which includes a centralized
administrative and pedagogical system of control. The school is located in the
neighborhood of the University and presents similar characteristics of other schools in the
city. In addition, this school is a place for prospective teachers to practice. It is a
medium—sized school with 1100 students, 50 teachers and an administrative staff. I spent
three months there performing participant observation. I had access to all facilities and
school activities. I was invited to participate in teachers’ meetings, celebrations, or other
events, in addition to everyday participation in the third grade mathematics class.

To collect data at school, I carried out participant observation in mathematics
classes and interviewed students, teachers, supervisors, the counselor, the secretary, and
the principal. Being at school every day, I could establish a good relationship with those
in the school. After some time, my presence there was taken as commonplace. I took
photographs and videotaped special events and some of the routines of the school day,
such as lunch and break time. I did not videotape any classes, because the use of the
camera could have disturbed the teacher’s routine in conducting classes. The camera
itself was very unusual for them. The use of a video-camera was not part of the daily
routine of that community of working class families, small scale vendors, and

consu'uction workers.

47

48

The teacher I worked with was an undergraduate student teacher with a major in
Language Arts. Linda, as I will call her from now on, is a young, active woman, very
interested in music, politics, and education. She comes from the school neighborhood,
and she knows each student as a neighbor. Linda has worked at this school for fifteen
years. Linda was very accepting, and made me feel “at home” with her. The only
concern Linda expressed to me when we first talked was that she did not have “different
things to show me such as different materials and new methodology” (Field notes:
August 6th, 1990). In the classroom, besides observing, I took field notes and
audiotaped each mathematics class. After some time, because Linda insisted on having
my active participation, I started helping her with some student activities. I usually
followed the students doing mathematical problems and correcting homework, and
explained to them when they came out with some doubt about Linda’s explanations of
the subject matter. In this period of my study, Linda was teaching division. My
participation in mathematics classes with the students was very important because I
could both establish a close relationship with them and learn about their ways of solving
mathematical problems in the classroom context.

In this section, I will present the school context where children’s experiences
involving mathematics occurred. I will first portray the school as the arena where
institutionalized experiences are planned to promote mathematical learning. Then I will
describe the classroom setting where those experiences might actually occur under the
supervision and guidance of the teacher. To do this, I will look for the routines, rules
and main activities that occur in class, and the relationship that involves teacher and
students. At the end, I will describe the actual activity of teaching, as I observed it, in a

class about division.

49

W

Within the school, the supervisors are responsible for planning the curriculum that
is followed by teachers in their teaching. The supervisors' plan summarizes the content
knowledge in each subject matter, its sequence, objectives, and suggested activities for
the academic year. Each semester, students are given tests prepared by teachers under
the supervisors’ guidance. The general approaches for the elaboration of the curriculum
are given by the city’s Secretary. The orientation is based on units of teaching where
certain phases should be followed in the teaching process. The first phase is an
exploratory one. The teacher should make sure that the student is “ready” to learn the
subject matter to be taught. Also, the teacher is urged to look for connections with
concrete experiences of the student related to the topic. The next three phases are
Presentation, Assimilation, and Organization. They correspond to the introduction of the
theme to be studied in the classroom, the activities and practice with the new content
knowledge, and the discovery of relations in order to organize what was learned. At the
end of the process is the evaluation of the objectives achieved.

The pedagogical orientation given for third grade mathematics is that the theme of
division comes after the teaching of the basic operations of addition, subtraction, and
multiplication. Also, the teaching of division should follow some steps, according to
levels of complexity in algorithmic procedures. The following is a summary of the
recommended steps:

1. Division of whole numbers involving basic facts of division.

Ex.: 24 -:- 3,18 + 2, 45 + 5, 30 -:- 6, and so on.

2. Division involving basic facts with remainder.

Ex.: 18 + 4, 45 + 6, 23 -:- 4, and so on.

3. Division of whole numbers without rcgrouping.

50

Ex.: 66 -:- 2, 63 -:- 3, 48 + 2, and so on.

4. Division of numbers with remainder without rcgrouping.
Ex.: 38 -:- 3, 47 + 4, 65 + 3, and so on.

5. Division of whole numbers with three digits.
Ex.: 369 + 3, 648 + 2, 468 + 2, and so on.

6. Division of numbers with three digits with remainder.

Ex.: 365 -t- 3, 645 + 2, 469 + 2, and so on.

Only one supervisor was responsible for the five, third grade teachers. Having to
work with the five teachers as a group was difficult. The supervisor told me one day that
she was having problems getting everybody together, because it was necessary to replace
the teachers to let them be available for a meeting. There were no extra persons
available at the school to give this support. In addition, everyone had other activities
after class. Therefore, to have meetings it was necessary to cancel classes. During the
three-month period that l was there, they met just once. Linda, the teacher I was working
with, told me once that she was willing to have those meetings, because she wanted to
discuss questions related to evaluation and content knowledge planned for her class.

I had the opportunity to observe one of the pedagogical meetings called Conselho
de Classe (Class Council). As the supervisor explained to me, the idea was to deal with
problems based on what each of the teachers knew about the specific students from
previous years of teaching. The meeting was supposed to be a time for teachers to
exchange personal experiences and knowledge in dealing with problematic students. I
was told that this meeting is held every year, twice each semester. The supervisor also
told me that they have tried to hold separate meetings of teachers for each grade. Finally
she told me that the best situation is one where teachers from different subject matters in
the same class get together, because every teacher is dealing with the same students.

Her perspective was that in this case, there shouldn’t be any problem keeping them

interested in the discussion. This was not the case in the elementary school, because

51

there every teacher was teaching a different group of students.

At the meeting I attended, all teachers who worked in the same shift i.c. before
noon were there. The principal, the school’s counselor, a social worker who attends the
school, and the supervisors were there, too. The first part of the meeting was a
presentation of slides by the social assistant, about human relations on the job. The
message was the importance of being a leader, regardless of the position you are in, and
the effectiveness of working together. At the end, none of the teachers wanted to make
any commentary about the sequence of slides.

The second part of the meeting was the group discussion of teachers in each grade.
I stayed with the group of third grade teachers to follow the discussions, but, as the
supervisor had predicted, the discussion was limited because each teacher had a different
group of students. I asked myself how teachers should feel about getting so little done
after so many hours of work. Reflecting on this and other events, I realized that the
school witnesses many teachers’ struggles. Some of them might remain hidden, whereas
others might surface within the routine of the school.

The surfacing I observed happened naturally through the teachers’ informal
conversations among themselves in the teachers' lounge, the patio or the corridors. The
following entry from my journal about my conversation with two elementary teachers

during a break is an example:

Yesterday, I was talking with Linda and the teacher responsible for the library
about the current situation of the teacher as professional. Both were upset with
the situation, saying that being a teacher is nor a good thing these days. Teachers
don’t have money to afford any leisure, they cannot give their children even
minimum things such as money to have lunch at school. They cannot buy good
clothes, have vacations, or even buy books that are important in their profession.
Linda told us that she bought two books last month and, because of that, she is
short of money and having problems. Linda used an expression that was sad but
expressed exactly how teachers see their profession. She said: “I stay in line
[waitin g to get paid] more time than my salary stays with me.” It means that they

52

stay in line for a long time to receive a small amount of money that they have to
use immediately and in full to pay everything they have to pay. (From my
Journal, September 28th,1990).

Some further information can give us a better sense of the professional conditions
under which teachers work. The initial salary for an elementary teacher, working from 7
am. through 11:30 am, without considering the extra time for official meetings,
planning, and paper work was approximately three times the minimum salary at the time
of my observations. However, when the high Brazilian rate of inflation was considered,
the result was a significant devaluation of salaries. For instance, an elementary teacher
who had a salary of $250 in the beginning of this study, was receiving only $170 three
months later, once devaluation was taken into account, even though she was getting
wage increases in local currency. Checking one’s wages against a stable currency is a
common comparison to assess its real value. Thus, the wages of this teacher were
shrinking as time passed.

I also had a sense of the teachers’ professional environment from other entries in
my journal:

In another situation, teachers having coffee in the lounge during a break discussed

when the problem about noise on the first floor. Students having lunch were caus-

ing the noise. A supervisor told the teachers that she was worried about this prob-

lem. Teachers working on the first floor could not work during this time because it
was too noisy. She said that teachers should look after their students, keeping them
all together as well as preventing them from playing during this time. Perhaps, she
added, it would be better if only students who have school lunch go out at this time.

The other teachers and students would continue the normal class. One of the

teachers said: “but, if a teacher wants to have lunch, she has the right to do this.”

The other supervisor jumped into the conversation, saying that it was not possible

to keep students inside the classroom without the teacher. Someone else suggested

53

changing the lunch time. Other teachers argued that, not having an exact time for
the lunch break could have bad implications for conducting classes. She pro-
ceeded, saying that teachers would not know if they had time to begin a new sub-
ject or to finish previous subject. As a result, she concluded, they would lose a lot
of teaching time. This discussion ended, with no solution to correct the noise

during lunch time. (From my journal, September 28th, 1990).

W

Linda's third grade classroom was also a classroom for other students in the
afternoon shift, and still other secondary students in the evenings. Simplicity was its
basic characteristic, in its fumiture, building construction, and appearance. Since the
school accommodated students in three sequential shifts with different students in each,
it was not uncommon to hear students complaining about things that were misplaced or
had even disappeared. There were not enough personnel to guard school property, or
even time between shifts to clean. There were 30 student desks, a blackboard, a
teacher’s desk, and a cupboard large enough for keeping the students’ essential materials
to be used in the class: books, notebooks, and chalk. What struck me the most was that
there was nothing on the walls, or anywhere else, that could give any identity to the
classroom or that would relate to the third graders.

When the school day started, the classroom was quickly transformed. There were
close relationships between the students and the teacher, and among the students. Linda
knew every one of the 30 students in the class. In our conversations, she talked to me
about them with affection and concern.

Carlos is serious..., pays attention, asks question when has doubts....

Maura is calm, quiet, seems not to be enjoying school but seems not to have diffi-

culties in learning.

William has some difficulties. His mother is concerned... it seems that he needs to

54

pay more attention...

Anita has a lot of difficulties in learning. She didn’t know her mother... Her father

is very concerned about her.

Linda was one of the few teachers who stayed with her class during break-time.
Almost every day I saw her playing with the students. After break she used to sing with
them in class, to calm them down before starting teaching again. At the end of the
school day, she always received hugs and kisses from them when it was time to go home.
The class routine started at seven in the morning, with a prayer that Linda used to say
with the whole class. All of her students were caught up in an attitude of respect
following her words, quiet and in silence.

Each day Linda corrected the homework from the day before. These were
opportunities to repeat some points she considered important and to clarify doubts about
any mistakes or student misunderstandings. Sometimes Linda called on students to solve
problems or make calculations on the blackboard. As she mentioned to me once, she
used this time to call on students she knew needed explanations, or the ones who could
help the others. In fact, as I observed she took care that, at some point, everybody had
an opportunity to do different things in the class.

After correcting the homework, Linda inuoduced the subject matter planned for the
day. She always tried to get the students to share their experiences, asking questions and
listening to them. After this, she explained the topic, using the blackboard to register the
main points. At the end, the blackboard reflected the sequence of the class. Linda used
drawings and examples to illustrate aspects of the content she was teaching. After her
explanation, the students would copy what she had written on the blackboard, do the
exercises she gave, and correct them using the same scheme as they did for homework
correction. While students were practicing the exercises, Linda and I went around to the
student’s desks, looking at what they were doing and explaining problems that emerged

from the exercises or from the problems to be solved. I found it interesting that when

55

students gave the right answer, they demanded that we grade their notebooks and leave a
signal of our appreciation, such as “Very Good!” or “OK!”, or “Congratulationsl”.
Sometimes Linda used rnimeographed activities to give more time in class, but this was
infrequent, because there was just one rnimeograph machine for all of the teachers to use.

After finishing practice, it was time to copy the homework for the next day. If
there was another class after this, the students would keep their materials for
mathematics, and take out what was necessary for the next class. If not, they would say
goodbye to us and go home.

There were no rules of behavior posted on the walls, but implicitly there were
several rules that the children should follow expressed by the teacher’s words:

1. Pay attention when the teacher is explaining.

2. Do not talk while the teacher is explaining.

3. Do not disturb your classmates.

4. Do not copy while the teacher is explaining.

5. Do your homework.

6. You must have your material to work.

7. You may sit anywhere if you are well behaved.

8. You should ask questions at any time if you have a problem.

Despite these implicit rules, things did not always go well. There were moments
when I observed a lot of tension between students and teacher, or between a student and
another student. I remember a particular event that happened with one student I

registered in my journal:

Fabio is an interesting case. Unfortunately he doesn’t disclose himself in his
journal as does Anita. It is hard to get to know him because there are many
moments he doesn’t speak about himself. In addition he seems to be very strict
with himself and with the others. If the teacher complains to him about his bad
behavior or because he is not correcting the exercises, he expresses anger in his
face and doesn’t talk for while. After some time, he comes back as if nothing had
happened. One day he began to sing in a loud voice while Linda was correcting

56

exercises. She asked him to stop singing because he was disrupting other
students, and to pay attention. He continued singing. Linda said if he didn’t stop
she would send him out of the classroom. He answered saying that she wouldn’t
do this with him. She did. He went out and stayed there for about one hour. I
was coming in when I saw him standing up outside of the classroom near the
door. The whole class was coming outside to have lunch, and he asked Linda if
he could come in. She told him he should ask his friends, because he had
disturbed their learning. They went down to the first floor to have lunch,
including Fabio. After lunch and the break time, when the whole class was back,
Fabio insisted that Linda let him come into the classroom. Linda said that he
could if he asked his classmates to excuse him for what he had done. (At this
time I was there audio recording.) He came to the front and asked everybody to
excuse him. His colleagues began to ask him many questions about the reasons
why he had done that, and why he liked to bother everyone. He answered saying
that he didn’t know exactly why. He feels nervous sometimes, or angry, and he
needed to shout or do something like he did. (My personal journal, October 10th
1990, p. 74.) '

I have to say that I felt very uncomfortable with what was going on. While Fabio
was standing up in front of the class, talking with his classmates the teacher was sitting
in the back of the classroom. I imagined that Fabio would feel very bad to beexposed
like that to his friends, but he didn’t manifest this feeling. However, I also observed that
Fabio felt nervous about talking about himself in school, especially if we were talking
about bad performance and grades. On the other hand, he talked freely about his job
after school. At this time, he was collecting esterco (cattle manure) and selling it to
people to use in their gardens. He prepared the soil, planted, and took care of it after the
planting.

Fabio was n0t an isolated case of tension in the classroom. In general, students
paid more attention in the beginning of an explanation, but after some time they would
start playing around and making noise. Sometimes, Linda would become angry with
them and would start talking in a loud voice to overcome the children’s noise.
Sometimes this would cause the atmosphere of the classroom to become heavy and
uncomfortable. I observed this happening most when students could not understand

what the teacher had explained to them, or when she spent too much time explaining and

57

they were not very interested in the explanation. I noted that students had a repertoire of
activities for these occasions. Some examples follow:

1. Students rub glue on their hands, leave it to dry and start peeling off the fine skin
produced by the glue.

2. Students build estalinhos (a folded paper) and make noise with them by doing a
brisk movement up and down. The device produces noise as the air is expelled out of the
folded sheet of paper. I should remark that an estalinho is an ingenious piece of work.
The paper must be folded in geometric forms so that some parts of it move, expelling the
air and making the desired noise.

3. Students get up and sharpen their pencils, ask their friends for a colored pencil, or
an eraser, or something else.

4. Students write notes to friends, making jokes or sending messages.

5. Students play with figurinhas (small pictures that come with bubble gum).

Linda would complain strongly only when these students’ actions reached a point
where other students would be disturbed by the noise. On several occasions, I observed
her attitude regarding these activities. She would not take many of them seriously
enough to consider that the students were breaking the rules of good behavior in class.
Rather, she continued with her class as if nothing had happened. Usually the student or
students involved in this action would eventually relax and go back to paying attention to
the class. As I observed it, her strategy of overlooking these actions many times was
effective in allowing students to relax, while assuring the continuity of the class.

Next, I will describe Linda’s teaching and show her students in action as she

teaches a class on the tOpic of division.

I A Divi i
It was before lunch when Linda started the lesson. She drew a line on the

blackboard, on the left side, wrote the date and the name of the class (Mathematics,

58

August 14th), and started reviewing some basic concepts about adding, subtracting, and
multiplying. She asked the students what each concept was about and how to perform
each related operation. Adding was identified as “grouping elements,” subtracting as
“taking out elements,” and multiplying as “making addition easier” because it showed
how many times a number had been added. Then, Linda said: "We’re gonna learn a new
operation today. It is division!"

What follows is a transcript of the main parts of the lesson in which Linda
introduces the idea of division and the algorithm to divide. One student’s participation
will be indicated by “St” followed by a number, (Stl, for example), and “Sts” will denote

various students in simultaneous participation

Inuoducing the Idea of Division:
Teacher: What does this word “division” remind you to do?
Stl: Subtract
St2: Diminish
St3: Diminish the price of the... (many students talking together).
Teacher: Division reminds you of subuaction?! (with surprise).
St: No... it’s to distinguish, to divide (Students keep talking together and making a
lot of noise.)
T: (Linda stops her teaching to complain about this). I’ll stop the class and I’ll
wait. When you decide, we can keep going... (Students quiet down and she contin-
ucs).
Teacher: R. (a student) remembered a word.... When I said division, she remem-
bered setting apart. What more do you (all students) remember?
Stl: To set apart...
St2: Division

St3: Basic facts (everybody talks together and Linda complains again).

59

Teacher: Shhh.... do you remember that I told you that we do things well, or we

don’t do anything?

Sts: Yes, yes...

Teacher: May I continue?

Sts: Yes, yes...

Teacher: R. told us.... I want to know what this word means to you. She told us

that division reminds her of setting apart. Have you heard this term... have you

used this word in some way? Do you?

Stl: Yes

St2: Me, too

Sts: (students agreed)

Teacher: How?

81: I remember... to divide...

St2: I know... I use that when I play soccer... to divide the team.

Teacher: Yes, that’s it. Very good. He (the student) remembered using the word.

He divided the team: some for one side and some for the other side. And you ?

Several students suggested different examples of division, as for example: dividing
a team to play queimada (a ball game), dividing a job, a salary, dividing a loaf of bread,
a lunch, a biscuit, and others. I was impressed by the students’ answers because they
were expressing many situations from their daily activities.

After some time Linda continued:

Teacher: That’s it. You see... if I ask... did you see? I asked what division was

and everybody... only R. said “set apart”... You see how we are doing division all

the time? I say... let’s divide a blackboard... (as she had done when the class

Started). We are dividing all the time...

Stl: Divide a head of the... (students laugh)

St2: Divide the cars...

Sts: (making noise)

Teacher: Shhh...

Teacher: Then, what would it be to divide?

Stl: Separate

St2: Divide

St3: Repartiar (This is a way to say “share” in Portuguese. The correct word is
repartir which means "share.")

Teacher: Ah! Here is a new word... Repartiar, share, separate...

Stl: Share food... one dish for each one.

Teacher: We divide the time, see... before lunch we’re gonna do this and after
lunch we’re gonna do that.

St2: Ah! Mrs. Linda, dividing a grain of rice... to cut in the middle...(students talk
for a while).

Teacher: We’re gonna start learning division... fiom the beginning. It will be
division without a remainder. Later on, we’ll see other kind of divisions, but only
later... now... (pause). The first thing we’ll try is to divide everything exactly.
For example, the set of fingers on both of my hands. How many fingers do I have?
Sts: Ten

Teacher: Are they equally divided on each hand?

Sts: Yes.

Teacher: Let’s suppose that there is a limit in my hand and (while she was talking

she drew on the blackboard).

Figure 5 A set.

61

This isaset.

This is a set. In this case what would be my hand here and there.

Figure 6 : A set and subsets.

Stl: Set and....

St2: Set

Teacher: Set and set?! Set and....

St3: subset

Teacher: 0k, and what is the name of the subset that has the same amount, look

(showing her hand)... unless a person has any deficiency... one, two, three, four,

five... (counting on each hand). There are five fingers on each hand. There are the

same number of....

Stl: Fingers.

Teacher: Fingers... elements. Then what is the name of the set with the same

number of elements.

Sts: E... e... (students try to remember the word "equipotent").

Stl: Equipotent.

Several examples were given using the same structure of drawing limits as I _
described above, but varying fiom fingers to flowers, stars, candies, and other things.
Also, Linda used different amounts: 24; 36, 12. Some students went to the blackboard
to do the division based on what she had done earlier, using the same drawing. Then it

was time for a break. When the students came back from the patio, they sang with Linda

62

as they usually do.

Linda restarted the lesson to relate division to multiplication.
Division and Multiplication

Linda drew set limits on the blackboard using the same scheme described earlier.
Using tampinhas (metal bottlecaps with which children like to play) she filled the whole

set as follows:

 

Figure 7 A set of tampinhas.
She proposed that the students divide the set of tampinhas into three equal subsets.
As the students spoke, she did the division on the blackboard. She took the tampinhas

one by one and transferred them to each subset.

 

Figure 8 Dividing tampinhas.

63

At the end, Linda said to the students:
Teacher. Ict’sseeifyoucanmakesomeconnections Therewere 12: wedividedthem; pull
them apart in three... how many elements do we have in each subset?

Sts: Four

Teacher: Can you make any connections? There were 12 divided by three, equal
to four. (Students were making some noise. They did not answer her about the
connections). She followed up the explanation by saying:
Teacher: Let’s see if you did perceive... I’ll write here two ways of representing
that:
12+3=4
(While she was writing, the students were repeating in chorus):
12+3=4u
Teacher: What does 12 mean? It is...
31,: Set
St2: Element
St3: Twelve?
Teacher: Set. Three are the...
St]: Elements
St2: Subsets
Teacher: Subsets. Look: one... two... three... subsets.
Teacher: And four, is what?
St: Elements
Teacher:... are the elements of the divided set. What can you get from this opera-
tion here? Can you connect this operation with another one? Everybody should
think. Pay attention to this: 12 + 3 = 4... everybody quiet, don’t talk now. Do you
remember something that...

St,: I know, I know...

64

Teacher: Shhh. You’re not to talk. You're to think! Look! We got exactly: four,

four, four - three equal subsets (stressing). 12 -:- 3 = 4. Observe carefully and see
if you can relate this with another thing we studied before... are you thinking?

Sts: Yes

Sn12+4=3

Teacher: Let’s see what R. has to say. What did you notice?

R: 4 x 3 = 12 (while she was saying this, Linda wrote the same on the blackboard

repeating after the student 4 x 3 = 12).

Teacher: Do you all agree?

Sts: Yes

In these sequences I described the teacher introducing the process of division and

relating it with multiplication. Linda emphasized the division with the idea of setting

apart equal number of elements from a whole set. The equal division sounded to me like

a requirement to do division. This was reinforced by the connection made with

multiplication. The ideas of division as sharing or separating, and the requirement of

equal part was not discussed after that fust moment of the class.

Linda gave a sequence to the class introducing the algorithm to do division.

The Algorithm

Linda explained that the use of drawings to divide flowers, stars or other things is

not useful when the number of elements is big. She gave the example of dividing 72

elements by two. This would take too much time, she said:

Teacher:... if we know that four times three is equal to 12 or three times four is

equal to 12, (...) another way we have to represent that 12 divided by 3 is: (and then

1 2 '3— structure used to

do division in many
Brazilian school:

she wrote on the board)

65

Because we know multiplication, we don’t need to draw or count anymore. We’re

gonna know that 12 divided by three will be...

Stl: Four (Linda wrote on the blackboard)

Teacher: Four. Why Four? Because four times 3...

St2: Twelve
12 I 3
12 4

Teacher: Then, here we’ll subtract. Are there any tampinhas left?
Sts: No (in chorus)

Teacher: We do that:

_12[_3_

124

00

(She repeated everything again based upon the algorithm.)

Teacher: Twelve divided by three is four because three times four is 12. Two
minus two is... one minus one....

Stl: Zero

Teacher: This is the remainder. So, are there any elements left?

Sr: No

After this, Linda named each term, writing on the board:

dividend—~12 30—— divisor
12 4+— quocient

00 o——— remainder

66

She read each term, she wrote it, and repeated its meaning.

Teacher: Dividend. Why? It is the one to be... divided.

St1: Divide by...

Teacher: Divisor... it is how many times we are dividing. Quotient, is how many

elements we’ve got in each subset, and the remainder is what is left.

At the end Linda wrote some notes for the class on the board and told the students
to copy them. It was a synthesis of what they had done before. The class ended. The
students had almost 60 minutes of mathematics class.

Looking through how Linda taught this class, I observed some important aspects
that I will highlight here.

First, the ways students expressed their ideas about division was impressive. They
were able to draw upon a good repertoire of meanings for dividing from concrete aspects
of their lives such as food, money, and games. However, these meanings that were
contextualized in the students’ experience outside of school were not absorbed in the class
context. The examples were quoted and forgotten a few minutes later. The sequence of the
class continued, and covered the topics for that day. One serious implication of this
disconnection is that students might not have understood the symbolism that the teacher
presented in referring to their experiences.

Second, the different meanings of division the teacher chose to discuss with the
students were squeezed into the scheme of separating a set into two or more subsets. The
students expressed at least three different ideas about division which were not taken into
consideration: set apart, share and to allor. I will talk more about these ideas later. In
addition to this, the scheme used did not resolve situations where division was done into
unequal subsets.

Third, when the teacher introduced the algorithm for doing division, it came as a flat
picture of the process of dividing. I am saying this because it was not presented as a symbolic
representation of a dynamic process. The teacher said that the algorithm would simplify the

67

mechanics of doing division, but even in this case, the simplification came through a new,
and much more complex, mechanism for doing division. Each step of the algorithm should
have been clarified for the students, in terms of what it means and why it should be done.
Later, I will explore some aspects and implications of this mechanization for developing the
students’ understanding of the process of doing division using the school algorithm.

Finally, I would like to go back to some words the teacher used in this class. She
said at some point:

you see how we are doing division all the time?

we are dividing all the time !
And a few minutes later she said:

We’re gonna start learning division... from the beginning.

Later on, we’ll see other kind of divisions, but only later... Now....

If division is something that students do “all the time, ” what exactly are they
going to learn? Perhaps this is one of the questions mathematics educators should be

asking when planning what to teach students.

in v 1

During the three months I spent collecting data for this study, I was able to spend
several weekends with students on the streets. On these occasions I spent most of the
time talking to them about their experiences and games, or at the stores talking to the
vendors and observing childrens buying goods. I was interested in knowing how the
students dealt with real-world situations where mathematical knowledge was required.
My assumption was that this setting was a privileged place to learn about the social
construction of mathematical knowledge. These situations were so rich in data to that I
began going to each bakery, market and store in the neighborhood after class. In
addition to the interviews and observations I went to several students’ houses to talk to

their parents and relatives. There I could learn more about the students’ routines in their

68

daily lives and experiences. I used journals as an instrument to establish a conversation
with students about themselves. The journal made it possible for me to get their own
descriptions of a variety of games and activities that they engage in when they are not at
school. It is important to remember here that the school day lasts only four hours.

The students’ routine, in general, could be described as having the following main
activities:

-going to school;

-playing at home, on the streets, or at a friend’s house;

-watching television;

-helping their parents at home or at work; and

-buying things in small stores, bakeries, or at the butcher shop.

Among these activities, watching television was the least frequently mentioned,
although it is general knowledge that almost every family in this community has a
television set The children did not emphasize that watching television was one of the
main activities for them either in our conversations or in their journals. When it was
mentioned, they noted that they enjoyed watching cartoons, soap operas, movies, and
variety shows. No mathematical activity related to television programs was mentioned.

On the other hand, the activities that children mentioned the most during our
conversations and in their journals were games and other group activities on the streets.
The most frequent activities that children play in the street are games. There is a variety
of them, each one rich in opportunity for mathematical reasoning. Many times the
children’s mathematical reasoning was expressed clearly during their explanations about
these games, and other times they were implicit in the situation. I will report several

games and other activities children described to me or those that I observed in loco.

69

Games

The games they enjoy playing the most are played with a ball. Queimada and Here
are Brazilian names for the most common ones. Playing queimada involves:
1. Definition of two symmetrical squares or rectangles to be the field where two teams
play.
2. Definition of a smaller area at each end of the field to be the place for the prisoners of

each team.

 

P team 1 team 2 P

 

 

 

 

 

 

P = Prison

Figure 9 Playing field for queimada

3. Composition of the teams: In defining the composition of the teams, children take into
consideration the number of childrens who want to play, how strong and agile each
player is, and the equitable division of strength between the two teams. When there is no
agreement about the definition of each team, childrens usually decide based on
adedanha, a method used to decide who will be the first to choose a player. According
to adedanha, the leaders from each team come together to decide who is going to be the
first by choosing “odd” or “even” and hiding both hands behind their back. After both
children have chosen and hidden their hands, they say “a... de... dannn... nha!” and he/
she must show a number of fingers, at the same time as the other leader. They add the

total number of fingers showed by both leaders and the winner is the one whose choice

70

(odd or even) matches the number of fingers. The winner chooses a player first, and
both leaders take turns at choosing until everyone has a team.

Adedanha is used in several activities and other games, such as Bete and Pique
among others. In this situation, what drew my attention was the fact that children
bargain about the pairs even after the decision has been made during the adedanha
dispute. They look for the best balance of strengths through negotiation among the
leaders about their players. But there are other methods children use to attain a balance
when defining the teams. When there is an odd number of childrens who want to play,
or, when an equitable division of the kids’ strengths is not possible, they apply the
system of getting one extra “life.” This means that the team having one less player than
the other team can be hit twice by the ball before having one player sent to the prison.
4. The game is scored by counting the number of times each team takes all

the opponents prisoner. The player is a prisoner when he/she is hit by the ball.
5. Definition of the winning team: the winner is the team with the higher score.

This game involves constructing two circles of a meter or less in diameter about 10
meters apart. Three short sticks are leaved together to form a target. Two pairs of
children from opposing teams who take turns at pitching and hitting the ball. When a
batter hits the ball he/she can run back and forth between circles to score points until the
other team gets the ball. Winning can be done in two ways - either by knocking down
the opponents' or reaching 100 points. Betriz describes the game in her journal with the
following pictures and words.

“ The game of queimada I can’t explain, and also bete, but with here I’ll try to

draw. ‘3
2
.4 3’ ‘31

Figure 10 Playing Here 1 (Student's drawing #1)

71

We draw that thing on the ground. On the people’s hands are pieces of bamboo to
hit the ball with. Inside the circle are three sticks that we arrange. This bat is for
hitting the ball. The other two people throw the ball to tear down the three sticks.
The ones that hold the bat are pairs. The ones that hold the ball are also pairs. The
ones that hold the ball throw it and then I hit it and if I hit it far I get 100. If] get
1001 win one game and also have to trade bats like this. I’ll draw it.

Figure 11 Playing Bete 2 (Student's drawing #2)

 

Then the two keep trading places until they get to 100. And then trade until we win

one game: like this:

s M

\

, v b...” *“-_. ‘1' \

Figure 12 Playing Bete 3 (Student's drawing #3)

Another very common game that children enjoy playing is called pique. The child

who has the pique must run after other children and catch them. There are several

72

variations of this game:

-hide-and-seek: When children hide themselves in several different places from
the person who has the pique. One particular point is defined as a neutral zone and the
children cannot be caught there. If one child is caught out of this zone, he or she holds
the pique and the game starts again.
pique-cola: When a child is caught, he or she has to stay in that same place until another
child frees him or her. While she or he is on hold, the child cannot run. When everybody
is on hold, the pique starts again with another child in charge.
pique—”freeze”: When a child is caught, he or she is “frozen” till another player frees
him or her. It differs from “pique-cola” because in this case the child cannot make any
movement or change his or her position. The child that is caught has to act as a statue.
-pique- flag: Played by two teams. Each team has a flag and has to defend it from the
opponent. The field is defined as shown in Figure 13 and the flag stands behind where

the players are:

 

[I I]

 

 

 

 

 

f: flag

Figure 13 Playing Flag

The team who gets the enemy’s flag first scores one point, and the winner is the
one who has more points at the end. The flag is only symbolically defined. It can be a
piece of anything such as a wood stick, for example. The players must be agile and run
faSt. In this situation and others, where it is necessary to define two teams and to
balance strengths, the whole process of decision-making described before takes place.

There are some table-games that children play with a friend, but they are not very

73

common in this community, except for checkers and cards. Several card games were
mentioned, and some of them deal with specific values assigned to each card. But in
general, they count the points they get one by one, and the winner is the one who gets
more points than the other.

Another very popular game is called bafinho. This is a game in which two
children bet and dispute stamps or cards organized in a pile on the floor. They do this by
cupping one of their hands and hitting the pile with it in an attempt to turn over as many
as cards as possible. Each player gets all cards that were turned over. If none, another
player gets the turn. Each time a different kind of stamp is the object of dispute among
children. It depends on the type that is most popular at a particular moment. The cards
come with a chewing gum and there is a variety of them in the market. When these data
were collected the most popular ones when these data were collected were stamps about
animals in the Brazilian wetland, and different monsters. This game is more common
among boys than girls, although not exclusively so. In general, they play until they get
the card they want, or until they lose everything. In each game, a player can win or lose
over a 100 cards. Sometimes children borrow some cards from their opponent in order
to continue playing.

Another individual activity very common among boys is to roll an area (hoop).
This is a kind of game where children put an area into motion with a stick. The area
might be made from metal, rubber or another material, and can be small or big. It
doesn’t matter. The important thing is to keep the area running under complete control
of the child who is rolling it. The more the child can do with the area without touching
it, the better he is considered. Children strive for complete control over the area. The

following figure illustrates how children play:

74

\C
/\

 

Figure 14: Playing Arco.
There are other ball games children play that are very popular in the Brazilian
context and well known in the United States: soccer and volleyball are two of them.
The play activity that girls engage in, almost exclusively, is play-home or play-
school, where in pairs or in groups they simulate a situation of house activity or school
activity. These activities are interesting because children perform as adults do when at
home or at school. The children described what they do when they are playing school.
Most of the time, their friends are their students and they are required to behave and
perform things as they do when they are in the real situation of a school day. In an
informal conversation, Carlos told me how he does this: "I played with Anita that I was
the teacher. I wrote on the blackboard, she copied and uied to solve it, and then I
explained to her how to do it."
Fernanda also talked about how she plays school:
I play school with my cousins... They have a blackboard I play like this: she
gives me some calculations to do and I solve them. I make divisions and so on. I
play alone, too. Sometimes I throw papers around on the floor while I am
explaining... Sometimes I complain to the students as if they were real students;
I call the roll, apply tests, cry out about the test results, and even tear the tests up.

Also, I have lots of books. I have mathematics books and I copy a lot of exercises
on division and multiplication from them.

The girls expressed this activity through drawings where mathematics were

 

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involved. Some of their drawings were inspired by books, and others created by
themselves. Children work with mathematics as it is at school. They work with
fundamental facts of basic operations, make calculations, and solve exercises, all within
the same pattern that they experience at school, correcting and evaluating themselves at
the end. In addition to this, the children clearly say that “they are playing school, having

mathematics class.”

This drawing by a child is an example:

 

 

 

 

 

 

Figure 15 Playing school (Note: The child has written "you are able. Try this

subtraction")

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Buying Things and Other Activities

Helping parents is another activity children perform as part of their daily routine.
There are a variety of things childrens do within this category that are particularly rich in
opportunities for mathematical reasoning, because of the nature of the activities they
perform. This is particularly true because, when children help their parents, they deal
with money in buying and making change, making measurements, estimating, and
counting. Regarding experiences in buying and receiving change, Fernanda told me in

our first interview:

When I went to buy a loaf of bread I had 50 cruzeiros and it was 16 for four
loaves of bread. Four loaves of bread... 16, so I counted. It was note of one each,
so 17, 18, 19, 20. There were four n0tes of one and three of 10, then it was 50.

Vera helps her parents buying groceries at the supermarket. She described how she

does it:

On the day before, I checked out the prices and using a calculator I figured out the
total. Then I went to the market and I verified whether the prices were higher than
before I bought the things I want. To verify if the change was correct, I counted
first the notes that are smaller and then the bigger ones.

One interesting aspect of the activity children do when they are buying things at the
bakery, bar or small stores is the way they deal with situations where any mistake is
found in the payment that the children make, or in the change they receive. Vendors are
quick to say that, in general, all children are very smart in dealing with these situations.

Vendors also say that children are smart in trying to get some money to buy things
for themselves. One strategy they use is to bargain the price. For instance, children may
ask vendors to cut off the price so they can pay less and have some money left to buy
goods they really want to buy. One of the vendors I interviewed told me:

Sometimes children ask me to give them the receipt including a candy. Another
thing they do is to check if the price in another store is more expensive and then

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they buy where it is less expensive, and tell their parents that they bought in the
other store where it was more expensive.

Several vendors reported that, in general, children do not check to see if the change
is right or not. They take the change and go out running. In contrast, children talk
among themselves about situations where mistakes are made by vendors, and in some
cases, how they take advantage of this. They reported some cases where they criticized
vendors and pointed out what happened and how it happened. During my interviews
with some of them, they explained to me what they did when they had to face this kind
of situation.

Fabio: I went to the bakery to buy milk and I paid with 100 cruzeiros. I didn’t
check the change. At home, when I was going out, I checked the money that I had

in my jacket and I saw that I got the wrong change. I had already more than 100
cruzeiros. I kept 50 for me and gave the rest to my mom.

Vera: The other day I went to make change with two notes of 50 cruzeiros;
instead the vendor gave me three notes of 50. I checked the change and I saw the
mistake. I told my father and he demanded me to go back and give the money to
the vendor.

Dealing with money and being responsible for doing things by themselves gives
these children a strong sense of the value of things. The following conversation among
three students and the interviewer, about 1000 cruzeiros, shows their understanding
(1000 cruzeiros was equal to one fifth of the minimum salary per month at the time the
data was collected, which was equivalent to 10 American dollars):

Fabio: 1000 cruzeiros is a lot of money!

Interv: What can we buy with 1000 cruzeiros?

Eduardo: I can buy many things. I can buy a tire for my bicycle.

Interv: Is it enough to buy a tire for the bicycle?

Eduardo: It will not be enough for the special tire (a more expensive one), but for

the regular type, it is.

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Interv: How much does the tire cost?

Fabio: It costs 90, perhaps a little more...

Eduardo: It might cost around 900.

Interv: 90 or 900?

Fabio: 90

Eduardo: The special tire costs 2700.

Interv: This bill (a 1000 cruzeiros bill) would be enough to buy it?

Fabio: No! Are you kidding!

Interv: How much do you need to add to that?

Fabio: More 2 thou....

Eduardo: 1700....

Fabio: I can do a lot of things with this money....

Eduardo: I can fix my bicycle, and give the change to my mother to buy food.
Fabio: My gosh! I could eat a lot with this amount of money. I would do a lot of
things...

Interv: How could you divide this amount?

Fabio: I would divide this money with them (other students who were participating
in the interview). We would have a lunch buying a loaf of bread with salami (a
variety of mortadella very appreciated by children there). It would be three loaves
of bread with salami and three Coca Colas (Coke).

Interv: Would it be enough?

Fabio: Enough? (stressing) More than enough!

I showed them a 10 cruzeiro bill and asked:

Interv: and this, how much can it buy?

Sts. Almost nothing!

Fabio: It doesn’t buy even a half of 100 grams of the salami.

William: It buys one gram.

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Interv: How much does the salami cost?

Eduardo: 100 gram? 35 cruzeiros.

Interv: If the salami costs 35, how much could you buy with 10 cruzeiros?

Fabio: You have to add 25 more...

Interv: Yes, but 10 cruzeiros can buy how much salami?

William: It is not enough even to buy the white part of the salami... (refening to

the small spots of fat that can be seen in a slice of mortadella).

Eduardo: It might be enough to buy around 20 gram.

Interv: Now, considering that you know how to spend the money, it’s only neces-

sary to get it....

Eduardo: Spend!? No way! Only next year. I broke my father’s thermos and I’ll

get some money only next year.

(exerpt from first student’s interviews, September 21, 1990).

Children go to stores to buy things for themselves such as candies, gum, and
snacks, but they go mainly to buy things for their parents, such as loaves of bread, milk,
cigaretts, and groceries. When children go to the store for their parents, they get money
from their parents, who usually tell them if there will be change or not and how much it
should be. When children were buying things for their parents, I observed that the
majority of them did not count how much they had in hand to pay the amount they owed.
They brought the money rolled up, and gave it to the vendor. The vendor checked it out
to see that it was enough and, when there was change, they rolled it up and gave it to the
child in the same way. If, for any reason, the amount of money was not enough, the
vendor sent a note to the parents telling them how much was missing.

In some cases it did not happen this way. This occurred when vendors were
dealing with older children around nine years of age. In these instances, the vendors told
the children how much was missing and asked them to tell their parents. Sometimes they

looked to see how many items the children brought up to buy, and sold them the amount

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that the money paid for. This happened most when the child was younger, seven years
old or less, and she or he was trying to buy some candy or something similar. Some
children buy for their parents using a notebook, where the vendor writes what the
children are buying and the specific price, instead of paying in cash. This system of
buying using a notebook is very common in small communities in this region. The
vendor writes down what is being sold, and the payment is made at the end of the week
or the month, depending on how the worker receives her/his salary.

In one weekend that I spent observing children buying, I saw several cases where
children paid more attention to the amount of money they were carrying when they were
buying things for themselves. They did this when they didn’t have one specific thing to
buy; instead, they had to balance how much money they had with the price of what they
wanted to buy. I saw many children coming to the store, looking around, asking for the
prices of many different things, and finally asking for one that matched with the amount
they had. The following entry in my journal illustrates this situation:

One child was buying loaves of bread for his parents. He came and asked the
vendor the price of 12 loaves of bread. It was 54 cruzeiros. Then he asked for 50
cruzeiros of bread (he had 50 cruzeiros in his hands). I asked him how many
loaves of bread he would get with 50 cruzeiros. He answered 10. I asked the
vendor the price of each loaf. The vendor told me that it was 4.50 cruzeiros. I
insisted with the child saying: If each loaf costs 4.50 cruzeiros and you have 50
cruzeiros will you be able to get 10 loaves? The child reaffirmed to me that he
would get 10. Then I asked if he would get any change for paying with 50
cruzeiros. The child answered that he wouldn’t get any change. He got the bread

and left the place without knowing how many loaves of bread he bought. After he
left the vendor told me he gave 12 loaves of bread to the child.

In another situation three children were buying for themselves. One of them came
into the bakery, looked around and asked the vendor for the price of one bottle of
soft drink. It was 30 cruzeiros. The child kept looking around for other products.
A few minutes later he asked the price of another product and finally ordered a
biscuit. He paid with two notes of 10 cruzeiros and waited for the change. After

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receiving the change he checked how much he got and asked for the price of
another product. Finally he bought a loaf of bread, kept the change he had got in
the new purchase and went out of the store.

The second child was with his mother. He spent time looking around to find
something to buy. He didn’t ask the price of any product. He chose what he wanted

after some negotiation with his mother.

The third child came and asked how much one kind of candy cost. It was two
cruzeiros each. Then the child took some money out of her pocket and began to
count: one... two... (notes of one cruzeiro)... one candy. Another note... another
candy... until the end of the notes. The vendor asked her: how much money is it?
She answered: a lot of money. The vendor counted the money and there were five
cruzeiros. The girl said to the vendor: give me five candies. The vendor replied:
but the price is two cruzeiros for each candy! The girl then asked him: how many
candies can I have? The vendor gave her three candies and she went out, keeping
the candies with her.

This same perception was expressed by a vendor where many children go to buy

small things such as pencils, erasers, marbles, and inexpensive toys. She said:

I think that children are smarter than adults. They bargain on the price, ask if they
can pay later... Today children are very smart and clever. They look the
merchandise over before buying it.... Sometimes they spend one week looking for
the best deal before buying. They compare the prices...

Besides going to the stores, children help their parents doing housework or working
with them in their jobs. It is intereSting to note how different the way children make
calculations is from school mathematics. Vera’s father owns a butcher shop and she
works there with her mother and her eight-year-old brother, a second grade student. I
went to the butcher shop to observe Vera working, but she was not there. I did, however,
interview her father and observe and talk to her brother, Pedro. He was helping his
father, receiving money from the customers and giving change when it was necessary. I
gave him some situation problems involving change, and he was able to solve almost

every one. The following dialogue between us shows part of his reasoning to figure out

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the customers’ change. To start, I asked him the price for 100 grams of ground meat. He
asked the same question to his father, who answered him that it was 28 cruzeiros. Then:

Interv: If I buy 100 grams and pay you with a 50 cruzeiro bill, how much will I get

for change?

Pedro: 22.

Interv: What did you do? How did you do it so quickly in your mind?

Pedro: I counted that 28... 29... 30. Then it is missing 20 and 2, 22.

Interv: If your father tells you to give 43 cruzeiros of change to the costumer and

he gives you a 100 bill, would you know how much he spent? How much will he

be paying you?

Pedro: 57.

Interv: How did you know that?

Pedro: Ah! It’s hard to tell you...

Interv: And, if you have to give him 25 cruzeiros of change for 100 cruzeiros, how

much had he spent?

Pedro: 75.

Interv: What did you think to get this conclusion?

Pedro: I did like this 20... 30... 40... 50... 60... 70. ‘Then I thought that it was

missing 5, so 75.

When I talked to Vera later, she told me that she was not making change any more
because she had made a mistake, giving two 500 cruzeiros bills instead of one.

At home, children help their parents doing housework such as washing clothes,
sweeping, cleaning the house and cooking. Many of them are responsible for babysitting
their younger brothers or sisters. Some of their reports show how they dealt with
situations where measurements are required:

Ana: My mom tells me what I should do. It has to be a lot of angu (a dish pre-

pared with com flour) because my brother eats a lot and me, too. I have to put four

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portions of flour. I take a spoon and fill it four times. If it is not enough I put

more....

About the water, I put more or less to the middle of the pan. The pan is big and

heavy! My stepmother fills it for me because I am afraid to spill the water.

To cook the rice, I put five cups into the pan. I bring the water to boil and add in

the rice later. My sister is who defines the amount of water, and I do the rest.

The activities and games described above demonstrate that there are many
situations involving children that require mathematical reasoning. My description
also shows that the children deal with these situations solving problems using

strategies that are not taught in school. I will explore these strategies in Chapter 5.

CHAPTER V

CHILDREN DOING MATHEMATICS

Man

To learn how students generally deal with mathematical situations, and particularly
with division, I collected data in several different situations. First, I collected data from
my observations in a mathematics classroom. There, I followed students doing
mathematics in situations planned by Linda, the teacher. I observed the students when
they were practicing exercises based on the teacher’s explanations, in quizzes and tests
they took, and in questions they asked in the classroom. In addition, I observed how they
expressed their ideas in mathematics through their journals.

Outside school, I collected data by observing students solving problems in loco.
The most significant part of the data, however, was collected in informal conversations
and interviews with students. These interviews were very rich in data because I could
explore in more detail the strategies students used to solve problems. Repeatedly, I
asked them "What do you think that makes you say that? Why?"

Two experienced teachers and I interviewed each student twice. We conducted the
first interview in the sixth week of classroom observation and the second interview
during the eleventh and twelfth weeks of classroom observation, at the end of the data
collection period. We conducted the interviews in a very friendly, open and informal
climate. I invited the students to participate and they were free to accept or not. Each
interview lasted about 40 minutes and was conducted in groups of two to four students.
To be interviewed, the students had to come back to school in the afternoon, after the
school day. Initially, I thought that this factor would limit the number of students who
would participate, because they would lose part of their free time, but this was not the

case. For the first interview, 24 of 30 students were present, and 26 were present for the

84

85
second interview. Only three students did not participate in any interview.

In bath interviews one of the objectives was to learn how students deal with
situations involving mathematical reasoning. However, the first interview was a more
exploratory one. The conversation flowed from whatever the student said at a given
moment. The strategy was to elicit evidence and a description of the student’s
experiences with mathematical reasoning outside school. As a result, the students
described a variety of problem situations they face daily. This was important, because I
used these situations to plan the problem-solving for the second interview. From these
interviews, I was able to identify a variety of mathematical problems. Then I classified
the problems from very easy to very difficult, according to the school mathematical
knowledge that was required to solve them and the familiarity of the situation for the
student. For example, problems involving division with a remainder and decimals were
considered difficult or very difficult, because students were working on division of
whole numbers in mathematics classes.

The following are some examples of the problems students worked with in the first
interview:

1. mm: How many grams of salami can be bought with 10 cruzeiros
(Brazilian money), if 100 grams cost 35 cruzeiros?

2. Difficult: How much does one kilo of carrots cost if I buy two kilos for 85
cruzeiros?

3. Regular; With 23 cruzeiros, how many pieces of gum can you buy if the price of one
piece is 5 cruzeiros?

4. Easy: How can you divide 10 cruzeiros among 3 friends?

5. My: With 5 cruzeiros in change to be divided equally between you both, how
much money will each one of you get?

Not every child had to solve all of the problems, but all of the children had an

opportunity to discuss a problem and possible solutions during the first interview.

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Different kinds of problems were objects of discussion. The situations in which they
had to deal with money were the most frequent ones: figuring out change, the cost of
vegetables, bread and candy, division and combination of bills, etc...

While the first interview was more exploratory, the second was directed toward
solving specific problems involving division. For the second interview, I planned eight
problem-solving situations. I based the situations on the results of the first interview,
and on observations of the students in stores, supermarkets, and playing in the street.
My interest was focused, among other things, on learning from them what their
reasoning was in solving division problems. The focus of the interview was not on the
solution itself, but, rather, on how and why the student arrived at a particular solution.
After each student response, we posed questions like, “What was your thinking in saying
that?”; “How do you know that is the answer?" or “Why have you done this or that?”
The students had access to a piece of paper and pencil and if they wished, they were able
to write, evaluate, or make drawings to find the solution to the problem. However, this
was not encouraged, because I was interested in letting them express themselves beyond
the limits of the mathematical algorithms they had learned. In addition, I was interested
in comparing students’ strategies for solving mathematical problems inside and outside
of the classroom context.

I applied these same problems in the classroom. On the last day of my classroom
participation, I asked the students to solve these same problems for me on a sheet of
paper. The problems had been rnimeographed, as Linda usually does, with some written
exercises and quizzes. In exarninin g their responses, it is important to remember that 26
of the 30 students had already solved these problems in the interview.

The following are the problems in the format I presented to the students in our
interviews and in the classroom:

1. If you have 50 cruzeiros, how many loaves of bread could you buy if each loaf of

bread costs 4.50?

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2. If you spend 45 cruzeiros to buy milk and pay with a 50 cruzeiro bill, how many
candies could you buy with the change?

3. Seven childrens want to play queimada. How could you divide the two teams?

4. You have two oranges to be divided with three friends. How much orange does each
one get?

5. To organize a collection of 150 figurinhas in two albums, how many figurinhas
should I put in each album?

6. There are 45 loaves of bread to be divided between two classes of first grade students.
How can the canteen woman do this division?

7. If you use six sheets of paper to cover four shelves, how many sheets of paper will be
necessary to cover just one shelf?

8. Ms. Luiza has 30 cups of milk to be divided among the third grade students. But
only 20 students came to school today. How can Ms. Luiza divide the cups of milk
equally among the students? How much will each student get?

All students had the opportunity to solve each problem inside and outside the
classroom. In this chapter I will analyze how students did mathematics inside and
outside of classroom context. I will describe the strategies students used to solve
division problems and identify possible sources of difficulties they encountered,
especially in situations in the classroom context where specific objectives are expected to

be achieved.

lvi 1 i h l C n x
In the classroom, the students received a mimeographed paper with the same eight
problems we had discussed in the interview. When I handed out the paper, I explained
that they could answer the problems in any way they knew. A student asked me if it was
necessary to use division, and I answered that it was not, that they should “solve the

problem in the way they knew how to..” I did not interfere in what they were doing

88
unless they asked me a question. The traditional students’ questions came right in the
beginning:” Do I need to use this space?" "May I use the blank space at the end?” I let
them make these decisions according to what they thought was the best way to do it. The

one point on which I insisted was that they should try to show the work done.

Approaching the Problems

Analyzing the results, I found some general procedures related to how students
approached the situation problems in the classroom. First, the students took the problems
as if they were fixed situations requiring the application of one specific mode of
calculation to find the answer. Almost immediately after I handed out the papers and
students started reading the problems, they asked several questions about what operation
they should use:

Nando: These problems are all in division?

Laura: ...is to divide?

Vanda: Is it division or multiplication?

Beto: ...Does it need two Operations? Is it only division?

Ana: ...what I’m gonna do is to subtract

Even when it was recognized as a “known situation, ” they resisted the idea of
solving the problem based on what they knew. Repeatedly they said: "But I didn’t learn
how to do this ....” or "We didn’t learn to do division with these numbers.” Observe that
they referred to “doing division with these numbers, ” rather than “this kind of division”
or “dividing in such way.”

The second aspect I found relevant when students approached the problem situation
is that they did a linear reading of the problem. They gave more importance to the words
used in the problem than to the actions implied in it. The following short dialogue with a
student illustrates this:

Elias: I don’t understand

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Interv: What do you not understand?

Elias: The question is here How much is
She read the question for me and added: What is the question? Where does it start?"

In another situation, a student asked me to explain the problem. I told her that she
should read the problem with attention, and look for the information the problem gave
her and the question that was asked. She immediately asked me, “Can it be subtraction?”

I also observed that, in some cases, when they seemed to understand a problem
situation, they kept a focus on calculations and the algorithm rather than on actions
implicit in the problem and their results. Tereza is one example. She told me that she
didn’t understand the problem. Then, I explained the problem to her emphasizing the
situation of buying a loaf of bread in the bakery as she usually does every day. At the
end she asked me “Is it necessary to do the calculations?" " Laura also said: "Mrs.
Maria, this is difficult. How am I gonna divide this thing here by this thing here?" (She
was showing me the numbers involved in the calculations.) Observe that she was
referring to the amounts involved in the operation described in the problem. But, the
focus was on the numbers alone not on the action suggested in the problem.

There are some discrepancies among the cases. They came out when students were
solving problems about candies (Problem #2) and choosing teams to play queimada.
(Problem #3). Roberto is one of them. He came to me and said:

".Mrs Maria, you see this problem here? If you have 45 cruzados 5 cruzeiros is
left, ok? How many candies is it possible to buy? I could buy a half of a candy, because
the price for each candy is 10.” Other students also made comments about the
impossibility of buying candies because their cost was more than five cruzeiros, and that
was the change they had.

A third aspect I observed is that students insisted on looking for confirmation of
their procedures all the time. Questions such as: “Is this correct?” “Is this done in this

way?” “Is this calculation correct?” came as long as the activity lasted. It seems to me

90

that the circumstances of being in the formal situation of a class works as a constraint on
the students’ decision-making. This idea is reinforced by the fact that students used the
same pattern of resolving problems based on what was required in mathematics classes.
Although I had left only a blank space after each problem for them to show their work,
the students divided it as they usually did with Linda. Students divided the blank space
into columns where they wrote the representation of the operation to be done, the
calculations done, and the answer. This is how they solve problems in mathematics
classes. I observed only three students making use of drawings to solve problems. It
seems that, in the minds of the students, the non-traditional way to solve mathematical
problems is not the recommended way to go in class.

The exception to following the traditional pattern of classroom way of doing things
was the case of answering some particular problems. Several students were able to give
answers that were completely unusual to their daily routine in mathematics classes.
Students answer problems following the question as it is proposed. In the case of
problem #2, for example, the question was “How many candies can you buy with the
change?” Usually, students answer this kind of question as follows “I can buy x candies
with the change.” Instead, these are some of the answers students gave:

-"1 could buy 1 loaf of bread with the change."

-"I could not buy anything."

-"It is missing 5 cruzeiros to buy one candy."

I also observed that students often based their answer to a problem on the result of
some calculation, without considering whether the answer found was reasonable or not.
This might explain partially why there are so many students who give completely absurd
answers to simple problems. The data I collected are rich in this regard. The following
case is particularly interesting. The problem asked how the canteen woman could divide
45 loaves of bread between two first grade classes.

Fatima solved the problem as follows:

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45+2=810 45 '2

4 810
05

The answer to the problem was "Poderia fazer esta diviscio dividindo 5 par 2, 4 par
2 que da em 810." (The division could be done dividing 5 by 2, 4 by 2 resulting in 810.)

Observe that she did not know how to divide 45 by 2 following the algorithm. She
probably knew that each digit should be divided (4 and 5), and she knew that the
algorithm was developed by subtracting on the left side, and she knew that division and
multiplication were related in some way. However, she probably did not know the
nature of the relationship between division and multiplication. (I saw her make the same
mistake in classroom activities.) Also, she could not relate the numbers on the left side
of the algorithm and its function. By the way she wrote the algorithm, it seems that she
did n0t connect what she had learned about place value to the mechanics of the division
algorithm. Finally, based on the calculations she had made, she wrote the answer

without considering that it was complete nonsense.

Solving the Problems

I have been describing the students’ approach to mathematical problems in the
classroom. Now I will describe some specific strategies they used to solve these
problems. I should remind the reader that all problems involved the process of division,
because this was the topic Linda was teaching at that time. She had already introduced
the long algorithm procedures for doing division with whole numbers and division with
remainder. She also had taught that division was related to multiplication, as described

in Chapter 4, and that this could be used as a way of checking the correctness of the

92
division.

Generally speaking, the students presented a limited range of strategies for solving
the problems. Basically, they chose one of the four operations they knew and applied its
algorithm as it was taught in mathematics classes. Although the problems involved
division, students used all four operations to solve them. However, in their applications
of basic operations, it was not clear to me whether the students had made sense out of the
problems. In some cases there was no sense at all. Only two students used drawings and
they only did so for problems #4, #7, and #8.

Adding. Adding parts to get the whole and then counting how many parts are
added could be one alternative for solving these problems. Based on the work showed
on her paper, Ruth used this strategy. In the problem about buying loaves of bread
(Problem #1), Ruth started adding the price of each loaf, 4.50, but stopped after the first
two. By basing her answer on the result of this calculation, she made the mistake of
saying two loaves of bread. This was wrong because the total amount of money was 50
cruzeiros, not 9 cruzeiros. Besides Ruth, five other students also added to find the
solution of this and other problems involving the ideas of partition and measurement.
However, the use of addition did not make any sense, because they added the part to the
whole. Fernanda is one example. She added 4.50 (the price of each loaf) to 50 (the total
amount of money available). In addition to this, there were mistakes in the application of
the algorithm. Two students did not take into consideration that they were adding
decimals and they did not pay attention to the use of the decimal point in their
calculations.

Emil—"E Doing successive subtractions makes sense in solving any of these
problems. The ones that involve the idea of measurement are easier to do. Since the
whole and the size of each part are defined, students could subtract one part successively
until everything was gone. In problem #1, 12 students took this option, apparently. I am

saying “apparently” because none of them presented this reasoning in its complete form.

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As was described in the first situation of addition, students did only the first step in
subtraction. All of them made mistakes in the use of the algorithm for subtraction. As
they had done before, they placed the numerals incorrectly. These mistakes in the use of
the algorithm led five of the Students to absurd answers, such as 50 cruzeiros buying 400
loaves of bread at 4.50 cruzeiros each.

There is one interesting aspect that bears consideration here, and this is the use of
subtraction in a different sequence than they used in mathematics classes. Linda had
taught that, in the representation of subtraction, the first term is the value that is bigger.
However, some students represented subtraction in the same way they say it orally.
They wrote 4.50 - 50 instead of 50 - 4.50.

For problems involving the idea of partition, the use of successive subtractions
could have been one alternative, too. Students could have estimated the size of each
part, and taken it successively from the whole, if it was an infinite set. Or, in the case of
a finite set, they could take units one by one from the whole until it was gone, as Linda
had done with the tompinhas, when she inuoduced division in mathematic class. But
students would have faced two additional difficulties. If the estimation did not work
well, they had to redistribute everything again, or make new adjustments. Or, in the
second situation, they might have made additional calculations to get the results. What I
observed in the students' work is that they did not make estimations in order to subtract.
My understanding is that the students resisted the inclusion of a new value in the
problem on which they were working. They tended instead to reason with the data they
had. This was the case with problems #7 and 8. In Problem #8, about 30 cups of milk
to be distributed among 20 students, the subtraction was 30 - 20 = 10. Students did not
go beyond this to figure out how 10 cups of milk could be distributed again among 20
students. The result was a wrong answer of 10 cups for each student. Only one student
said one cup for each child.

In the problem about six sheets of paper to cover four shelves (Problem #7), the

94
student subtracted shelves from paper. She did 6 - 4 and answered 2 sheets. It seems
that she did not make any sense of the problem situation.

Myltjijing. This is the one of the most difficult strategies to apply in solving
division problems. It implies that the student has to estimate the size or the amount of
each unit to be multiplied until one gets the whole set. When the problem has the unit
defined, students have to estimate the number of times to get the whole set. In this case,
the difficulties I mentioned before for subtracting can also be applied here. But, it seems
to me that the major difficulty is the introduction of a new element in the problem
situation, defined in classroom context. Not one of the six students who used
multiplication made estimations. All 10 times that multiplication was used, I could not
make any sense of their strategies. They multiplied seven children by two teams, 150
figurinhas by two albums, or 30 cups of milk by 20 children. In this context, the
absurdity of answers was the common point among the students.

Dividing. Linda was teaching division at the same time I was collecting data. It
was therefore natural that the students looked at the problems thinking about the division
they were studying. The challenge, however, was to apply what they had learned about
the process of dividing, to new steps in the application of the basic algorithm. Following
the algorithm, as Linda had taught it in mathematics classes, the students were able to
solve only the problem about division of two teams to play queimada. Seventeen
students (63%) out of 27 solved this problem using the algorithm correctly. But this was
the easiest problem they had to solve. Only two students could solve more difficult
problems (Problems #6 and 7) using the algorithm they had learned in the classroom.
Based on these results, it seems that the students did not make sense of the school
algorithm for solving division problems. In spite of this, I found four students who were
able to solve the problem using their own version of the algorithm. All four students
took the total number and did the division in only one step. This is one example: 45

loaves of bread divided by two classes:

95

44 22
01

Some students uied to use the same scheme Linda had taught in mathematics class,

but they made mistakes such as:

45 | 2 _ so I 4.50
4 23 50 1.00

05 0°

01'

5
0

In these examples the students seemed to have no idea what the numbers on the left
side meant, and how to get them. They probably knew that there was some relation to
the number called the dividend. Also, in their prior practice, they had done a lot of
matching cases where the numbers on the left side repeated the dividend. In the second
example above, the student gave a partially correct answer. My guess is that sheknew
the result beforehand, at least partially, and then she made up the algorithm just to satisfy
the condition of having an algorithm for division in the context of the classroom. I had
the opportunity to observe this same student in other situations in mathematics class and
outside school. She revealed herself as being a very smart girl with good initiative on
solving problems. Except for this, the majority of the students could not solve the

problem based on the application of the algorithm taught in mathematics class.

96
l 'n i 1 m n x

Analyzing what the students did to solve division problems, I identified some
general procedures in the student’s strategies.
1. Rounding numbers to be worked in the problem.
2. Estimation of possible results when the answer was not easily found.
3. Simplifying calculations, particularly when the mathematical algorithm suggested by
the problem situation was unknown to the student.
4. Use of some mathematical properties of operations to make calculations easier, such
as decomposing big numbers into smaller numbers in order to deal with them.

To exemplify the above procedures and to present a variety of strategies students
used to solve division problems, some situations will be described here as they came out
during the interviews. The described situations refer only to the second interview, where

the eight selected problems were presented to the students.

Situation 1: Buying

In this situation, there are two types of problems. The first type is a problem where
a certain amount is defined in terms of cost. An example of this is the following
problem: “If you have 50 cruzeiros, how many loaves of bread costing 4.50 each could
you buy?”

The second type is a problem where the student has to find the available amount
from previous expenses made to buy something else. Example: “If you spend 45
cruzeiros buying a liter of milk and you pay with a 50 cruzeiro note how much candy
could you buy?” Note that on this particular problem, the cost of the candy is not
defined purposely. I was interested in knowing how the students would react to this
missing data in the problem. In this regard, some of the children asked how much the
candy would cost, but the majority of them assumed different values, according to their

own experience buying candy at stores near their houses. In this situation of “buying,”

97
students used a variety of strategies based on their own experiences that I will describe in

this chapter.

Situation 2: Making distributions

In this situation, there are two types of problems: 1) the ones dealing with
discontinuing division, as for example dividing people in two teams, and 2) the problems
dealing with division of a continuum such as dividing oranges and loaves of bread and
milk. I was intereSted in knowing how children dealt with these different situations and
if, in their reasoning, they perceived them differently. None of the 27 students showed
evidence of having a problem dividing under either of these circumstances. They
approached the problems naturally, both in situations with continuous and discontinuous
quantities involved. The students took this difference into consideration when they were
solving the problem and looking for the answer. In the strategies I will describe next, I
will identify how it ocurred and whether it affected the way they proceeded. Generally
speaking, in problems involving the situation of making distributions, two main
strategies could be identified: 1) using the remainder to balance the division, and

decomposing numbers.

Situation 3: Making measurements

Two problems were included in this category:
1. If you need six sheets of paper to cover four shelves in a cupboard, how much paper
is necessary to cover only one shelf?
2. Mrs. Luiza will distribute 30 cups of milk among the students of the third grade, but
only 20 students came to class today. How can Mrs. Luiza distribute equally the amount
of milk among all 20 students? How much milk will each student get?

Defining this as making measurements does not imply that these problems involve

the idea of measurement as defined in some of the Brazilian textbooks about division in

98
elementary school (Amaral and Castilho, 1990). This is to look for how many times a
given quantity is contained in a larger quantity, or how many times a subset is contained
in a larger set. The situation of making measurements here means that the children were
not necessarily counting to solve the problem. Their attitude in approaching these
problems, revealed by their answers, was related to size (Problem #1) or volume
(Problem #2).

In the first problem, the size of the shelves is a fixed determinant for the solution.
This means that the size of the shelf makes six sheets of paper necessary to cover four of
them. The childrens had to find out how much paper was necessary to cover one shelf.
The amount for one shelf was already determined when 6 was defined as necessary for
the whole set.

In the second problem, the children had to find out how much (volume) milk one
child could have if 30 cups were available for 20 children. In other words, they had to
estimate the volume of milk, in cups, each child could get out of 30 cups. For this set of
problems, the strategies the students presented were: 1) making successive

approximations, 2) proportion, and 3) decomposing numbers.

Estimating from their own experience

In this section, I will describe the strategies I found in the students’ solutions to the
problems we discussed in the interview. This strategy could be seen in two senses. First
of all, it was clear to me that students relied on their own experiences when they
approached the problem. I took a lot of criticism from them about some of the prices I
gave them. For example, they corrected me on different occasions because of the high
inflation rate we have in Brazil; prices are increasing all the time. From the beginning
to the end of my data collection, the price of a liter of milk and a loaf of bread had
changed three times. The following student’s answer about buying candy with a 5

cruzeiro bill illustrates how the approach they make to the problem is related to their

99

own experience:

I thought: 50 minus 45 remains five. Then, I ask how much the candy costs. If the
vendor say 2.50, I can buy two because 2.50 with 2.50 is five. If the candy is 1.00
cruzeiro, I can buy five because five times one is five. If the candy costs 2.00
cruzeiros, it is enough for two candies because adding two and two is four, and
four plus one is five.

Note that the price for each candy was not defined in the problem. The student
aswered the price she knew based or her own experience on buying candies in the
markets. Also she took into consideration that the price was changeable, because of the
current inflation.

In another sense, some students also relied on their own experience to reason about
the situation problems. This was not always clear for many of the students, but the cases
of William and Laura might illustrate how this happened. The problem we were
discussing was about how many loaves of bread costing 4.50 each could be bought with
50.00 cruzeiros. William explained that he usually buys 10 loaves of bread for 40.00
cruzeiros. If the bread was . 50 more expensive and he had 50.00 cruzeiros, he
concluded that he could buy 12 loaves of bread. His reasoning could be described as
follows:

40 cruzeiros buy 10 loaves of bread

adding 10 cruzeiros buy two loaves of bread

50 cruzeiros might buy 12 loaves of bread

Thus, his answer was not correct, but he didn’t give an absurd answer like the

100

others had when they were trying to use only algorithms they had learned at school.
Laura, for example, answered that she could buy 220 loaves of bread. She concluded the

following:

4.50 I 5

4 220

4.50 + 5 = 220 by doing: ';

5
0

However, when Laura was interviewed, she answered almost immediately 11
loaves of bread. During the interview she explained what she had in mind. She

said:

"I didn’t think anything, I did it from memory because my grandmother buys 12
loaves of bread every day and she pays 50 cruzeiros for them...
Other examples can be found in this chapter in different situations I will be

describing.

Decomposing Numbers

This strategy was very helpful and utilized often by the students in solving
mathematical problems. Basically, it appeared in almost all situations. It is interesting to
note that in the situation of making distributions and sharing, the use of decomposing
numbers differs from the situation of measurement problems. The main difference is
that for the latter, students used it mainly for establishing proportions in terms of prices
and quantities, and for the former the students used decomposition of numbers to regroup

operations to make the calculations easier, using a distributive property of the division.

101

Here, they made successive divisions with smaller whole numbers than the one given in
the problem, and then they made successive additions to find the exact answer. In my
interpretation, this process involved many mental calculations that were not expressed
verbally in the students’ explanations. In addition to this, the students had to keep track
of the partial results of the intermediary divisions in order to sum them up and find the
final response. The following examples show some childrens making use of this strategy
to solve the next problem:

There are 45 loaves of bread to be divided among two classes of first grade

students. How could this division be done?

When the interviewer finished stating the problem, Marco answered 22.5 almost
immediately, and he explained:
40 divided by 2 is 20 and 5 divided by 2 is 2 and one half.

His reasoning could be described as follows:

45+2
\ /
V

/’°’\

40+2 5+2

l l

20 2 and one half
adding

22 and one half

The solid line shows what the student verbalized about his reasoning. The dotted
line shows what might be implicit, but was not expressed by the child in his explanation.
Or, it could be expressed like this:
45 -:- 2 =
=(40 + 2) + (5 + 2)

102

= 20 + 2.5
= 22.5
To solve the same problem, another student uied to apply the algorithm he had
learned in the classroom. When he was interviewed, he had learned the algorithm only

for whole numbers without remainders. The result was what follows:

45 ii.
22/ 1/2

And he explained to the interviewer: "Because 4 divided by 2 is 2; now 5 divided
by 2 is 2 and a half plus 2 and a half 22 and a half."

A detail from my notes is important in order to understand what happened:
"Roberto was initially a bit confused when he was explaining his thinking to me about
how to put together the number two, result of the first division with two and a half, result
of the second division."

To me, it seems that he was confused because the way he was supposed to perform

using the school algorithms did not fit his own way of thinking in solving the problem.

l—-4;2=2— — — -

I

He might have thought:

 

l— — +20 = 22.5

Another way to express this follows:
45 -:- 2 =
=(40+5) -:- 2
=(40 + 2)+(5 -:- 2)

103

=20 + 2.5 = 22.5
Another example could illustrate this same situation, where the student seems to
use the school algorithms to start out, but mixes these up with his/her own way of
thinking. It comes from Edina. I gave her the same problem and she said:
Edina: 45? (silence)
Interv: Tell me what you are thinking...
Edina: I thought like this: 45 divided by 2 is two times 2 equal 4, two times 2
equal 4, then I have 22. Two times 2, 4. 45 less 44 is equal 1.
Interv: And then, what conclusion can you make?
Edina: Each class will get 22 loaves of bread and there will remain one.
She described the first steps according to the algorithms she had learned in
mathematics class:
Step 1 : 45 |2_ “45 divided by 2”
She had learned how to divide whole numbers and do basic division calculations.
This means that doing divisions involving tens and units were not part of the students’
skills taught in math classes yet.
Step 2 : 45 |_2_ “is 2 times 2 equal 4, 2 times 2 equal 4”
4 2
At this point, the next step would be subtract four tens and divide five units like this:
45 |2_
- 4 22

05

104

But the student took the other way:

...2 times 2 is 4, then I have 22. 2 times 2.4
45 less 44 is equal 1 or 45 I 2

.14. 22
1
It seems that the student's mental calculation was:

454-2

40 5

2 times V2 times 2

22

!.

45-44=l

In this case another aspect should be highlighted, and this is how the student dealt
with multiplication to solve the problem. It seems that she did not use multiplication as
repetitive additions. It seems that the idea of multiplication embedded in this reasoning
is to look for how many times a specific number is within the other one. Thus, 22 is two
times in 44, and two is two times in four.

In the problem of organizing 150 figurinhas. into two albums, I identified several
examples of the strategy of decomposing numbers: Silvia explained how she could
divide 150 into two albums: "I could do it like this: 50 for each, it would remain 50.
Then, 20 for each and it would remain 10. And I divide five for each, it would be 75 for

each one."

105
It could be expressed in this way:

15}+\ 2
50 50
\ o
\ remains
add \ 50
\
\ /\
\ \ 20 20
add \ remains
\
\ 10
\
\
\ 5 5

Or it could be expressed like this:
150 + 2:

=(100 + 2) + (50 + 2)

=50+20+(10-:-2)

=70+5

= 75

Another example comes from Fatima, who gave the answer for this problem in 15

seconds. She explained her reasoning: "In 50 don’t I divide 25 and 25? I took 50 from
100 and put 25 and 50."

106

According to what the student said, the following scheme could be illustrative:

/ \
\
100’ '50
/
I \ /\
/
50 50 25 2?
l I I
I I
I l
75
r_____.______.'
Or:
150 -:— 2 =
=(100 -:- 50) + (50 + 2)
= 50 + 25
= 75
Another student, took less than 5 seconds to answer this problem. He explained his
reasoning:
100 divided by 2 is 50 and 50 divided by 2 is 25.
Or:

150/-'—\ 2
/ \
/ \
\
1002 2 + 5? -2
50 + 25 = 75

The following examples are interesting, although the students were not able to find the

correct answer by themselves. This is one case: "It is 53 because 200 divided by four
is equal to 20 no, I mean 50, then 25 divided by two, I will have 55."

107

In trying to capture what is implicit in this expressed reasoning, some questions
came out: Why did the student divide 200 by four? Why did he work with 200 instead
of 150? Was dividing 25 by two a mistake? Or was he, in fact, dividing 50 by two and
he made the mistake when he divided it twice? There are several possible answers to
these questions, and to check each one would require further interviews with the student.
This was impossible within the constraints of this study, but one hypothesis of
explanation should be explored here. The student rounded the number to facilitate his
calculations. In this study, rounding has been one common starting place to set up
strategies for solving several problems.

Here, 150 could have been rounded to 200. Then, 200 was divided by four and
not by two, possibly to compensate for the earlier rounding. The student probably chose
four and not three because it would be easier for him to divide 200 by 4 instead of 200
by 3. He might have known that 200 divided by four is equal 50. In several informal
conversations I had with some students, I observed them making this kind of calculation just
to show that they knew it Then, a possible representation could be:

50 for album one and 50 for album two

50 + 2 = 25 for each and 50 he left out
With SOflgurinhas left, he had to divide again among the two albums. So, 50 divided
by two is equal t025. He might have made a mistake in saying 55.

The second example is from a student about 11 years old. She started her studies
two years later than the usual school-age, that is seven in Brazil. During the interview,
in solving the same problem about the figurinhas, she worked all the time with
combinations of numbers to get the total of figurinhas she had or should have. But the
interesting thing about this case was her refusal to accept that it was possible to divide
150 by two, being fair. She said that each album could have 50 figurinhas, but the 50
remainders could not be relocated because one album would have more figurinhas than

the other one. She offered the following alternatives:

108

a) 50 figurinhascould be kept
b) 50figurinhas could be taken out
c) 10figurinhas more should be bought to allow 80 on each album.
d) to buy a third album and put 50figurinhas on each one.
It seems that her difficulty was in finding out that 50 divided by two is 25. In the
problem about 45 loaves of bread divided among two classes, this same student could not
divide five by two. Her refusal was so strong in both cases that I started thinking about
some particular difficulties she could have that would cause her to avoid these divisions.
I raised two hypotheses as explanations. One possibility is that Anita was dealing only
with whole number multiples of 10. She made combinations in such a way that she
could operate easily, as for example:

30 and 20; or

20 and 20; or

30 and 20

To deal with multiples of 10 would be easier, because she could eliminate the zero
and deal with numbers from zero to nine. In this case, dividing 50 by two would imply
solving 5 + 2 and transferring the result 2.5 to 25, because she should be dealing with 50,
not five. The constraint of having in mind whole numbers from one to 10 would make
this transference difficult. However, this same student easily solved the problem about
dividing 30 cups of milk among 20 children. She answered one and one-half cups for
each child. She could not explain what she had done in her mind to find the solution.
The difference between the two situations was that in the problem about figurinhas, the
division was with discontinuous quantities. In the division with cups of milk, there was a
continuum. This could be the reason for her difficulty in the previous problems.
Another possible explanation is that she could have been thinking in terms of the

formal division she had learned in the classroom, where decimals had not been

109
introduced yet. Neither of these hypotheses could be verified, because the subjects of
this study were not available after some time.

Finally, I would like to present an example of the decomposing numbers strategy
within the situation of buying candies with change of five cruzeiros after paying 45
cruzeiros with a 50 cruzeiros bill (Problem #2). This case is particularly interesting
because the student worked in two directions: going from 50 cruzeiros he had initially,
and coming back to 45 cruzeiros he would need to pay for a liter of milk. The dialogue
with the student follows:

Eduardo: I can buy two candies costing 2.50 each.

Interv: What did you think to conclude this?

Eduardo: 50 minus 2.50 is enough for two candies...

Interv: 50 minus...

Eduardo:...five, then I have 45.

45...50 minus 2... 50 minus... 2.50 will be 7.50...

Interv: Eh...take it easy! (sounding like I was not following his explanation) You’ve

paid with a bill of...

Eduardo: 50.

Interv: ...To pay for how much?

Eduardo: 45 for the milk and then, I’m gonna...10... 50 (he probably was looking

for the amount to complete 50 cruzeiros) and, with the five that remains from all, 50

minus 2.50 will be 7.50...47.50 minus 2.50 will be 45.

He previously had 50 cruzeiros, and he knew that he could spend money until 45
cruzeiros remained, since this was the price of milk. He knew that five was left to buy
candy. However, how could he get 7.50 out of 50 subtracting 2.50? It would be possible if
he was thinking about 10, as he initially started to say, and not the whole 50. In this case,

110

one possible graphic representation could be:

 

§O\ - 45 l
/ \
40’ 10 - 2.5 2'5
7L — — — 47.5

Or: 50 - 2.50 And: 47.50-2.50=

=(40+10)- 2.50 =(40.00+7.50)-2.50

=40+(10-2.50) =40.00+(7.50-2.50)

=40+7.50 =40.00 + 5.00

=47.50 =45.00

Another possible explanation is that he knew the results and, when he was asked to

explain his answer he tried to go back, “proving’he was right.

Using proportion and equivalence

Based on the data collected during the interview, I found some particular cases of
students using proportion and equivalence. This happened in the situation of making
measurement and, possibly, when they were reasoning about buying. Some children
found the correct answer for the problem by establishing relations of proportion or
equivalence among the data. One case was based on a drawing. This strategy was not
used in formal terms as someone would do in mathematics class in higher levels. It was
done based on concrete and simple situations of the everyday experiences students have

outside school.

111

To solve the problem about 30 cups of milk to be divided among 20 students,
Eduardo drew 30 cups. He put a mark point after each three cups while he was counting
“three for one, for two, for three childrens until the tenth child. Immediately he realized
that the answer was 1.5 cups for each child. I asked him what he had done to find out
that the answer was not three cups for each child, but one-and-a-half per child. He
answered: "With three by three there was not enough... it was 10 One-and-a-half
plus one-andoa-half is three, as I had done. By three I got 10 children, since 1.50 is a half
of three I could get 20."

I observed that Eduardo worked with counting by multiples initially, but he
realized that he was wrong when he perceived at least two things. First of all, the
relationship between the number of cups of milk and the number of children - more of
cups for each child, less children can get milk. Second, the proportional relationship
between three and one-and—a-half cups and 10 and 20 children. It could be described as
follows:

3 cups for each child ....... 10 children ........ 30 cups
1.5 cups for each child ..... 20 children ........ 30 cups

It could be expressed as follows.

 

 

3 cups , 30 cups th 1.5 cups _ 15 cups X CUPS
—— en ' ——
1 child ' 10 children 1 child ' 10 children ' 20 children

Vera solved the same problem in a simpler way. She immediately knew that every
child could have one cup and 10 cups would remain to be redistributed. To solve the
problem, she related one cup for two children then, proportionally 10 cups for 20

children. The representation could be:

lcup , 10 cups
2 children ' 20 children

 

The same thing happened when I gave this problem to a group of three students.

1 12
They all had the same reaction. They knew that each child could receive one cup of
milk and 10 cups would remain. The major problem then became to find out how to
divide 10 cups among 20 children. How they solved this part of the problem is
described in the following conversation:

Interv: What can I do with these 10 cups?

Laura: Cut them in half.

Interv: Cutting them in half, the milk will be enough for everybody?

Roberto: Only if it is one-half for one and one-half for the other.

Laura: One-and-a-half for each one because there are 10 and there are 20 children,

then one cup and a half for each.

Roberto: It is only a half because the remainder is only 10 cups, dividing by

20...(He had understood that Laura was saying one-and-half for each child in the

second division.)

Laura: That’s it. Divide one cup in two and it will be one-cup-and-a-half .

I reminded them that Laura was referring to the total that each child could get, because
they had already divided one for each child before. And the conversation followed:

Interv: How have you thought to find out the answer of one-cup-and-a-half?

Roberto: Idon’t know.

Interv: You knew that 10 cups were excess.

Roberto: That’s it, 10.

Interv: There were 20 children.

There are several interesting aspects to consider in this dialogue. But, for the
purpose of exemplifying the strategy they used to solve the problem, I will highlight two
points in the conversation:

I. The relationship was established when the child expressed that: "If 20 is the
double of 10, has to be half to be right.” It could be written this way:
"If 20 is the double of 10, then 10 is half of 20."

113

 

or one cup half a cup
20 children ' 10 children

2. The relationship was established by Laura when she explained that if there were 10
cups and 20 children it would be necessary to divide one cup in two. And then each

child could get 1.5 cup of milk. Or:

 

10 cups , one cup half a cup
20 children ' two children ° one child

In the following examples, I could not distinguish clearly whether the student was
thinking in terms of proportional amounts or if he/she was making successive additions.
This doubt exists because several amounts were mentioned by the student and the
interpretation may vary according to which amount is considered as the one which
guided the student’s reasoning. The problem that we were discussing was about buying
candies with five cruzeiros of change (Problem #2). Eliana said that she could buy two
candies and she would have one cruzeiro left because:

" .. each candy costs two cruzeiros. Two plus two equals four, so... I can buy two
candies; four plus two I can buy three candies, four plus four equals eight, so
there’s one cruzeiro left. I’ve paid four-and-a-half for the milk, and there’s five left;
so I buy two candies and there’s one cruzeiro left.”

Another way to say the above could be:
Each candy costs two cruzeiros;
2 + 2 = 4 buy 2 candies
4+2: 6 buy 3candies

4 + 4 = 8 buy 4 candies.

114

Or, if thought in terms of proportion, it could be described as follows:

2 cruzeiros 4 crizeiros 6 cruzeiros 8 cruzeiros

l candy ' 2 candies ' 3 candies ' 4 candies

My interpretation is that, if the student was adding successively, she probably
would say:

2+2: 4 buy 2candies

4+2: 6 buy 3candies

6+2: 8 buy 4candies

But instead, she might have realized that she could simplify her calculations by

relating the amount of candy she could buy with the amount of money she was spending.

Then:
2 cruzeiros 4 crizeiros 8 cruzeiros
l candy ' 2 candies ' 4 candies

Another possibility is when she got to 4 + 2, she added the next 2 to the old 2, and
got 2 fours. Then:

2 + 2 = 4 (2)

4 + 2 = 6 (3)

4+(2+2)=4+4=8(4)

Another very interesting example is the one where the student tried to figure out the
answer to the same problem by making the balance of different amounts. In this
situation, the student had spent 36 cruzeiros buying milk with a 50 cruzeiro bill. He

explained how he found out that the change would be 14 cruzeiros:

l 15
"1 divided like this: it is not 20, if so, it would be 24, and I would have 64. It

would be... I put 14 because I already had 30, so it would be 36... Then I decided
for 14."

In trying to understand what he thought in order to say 14, at least two possibilities

come out, and I will explain them using the following diagram:

20+14-24
ll 36 '=" 40

30+6 - 36
so >50
" 10+4-14

Here, the student rounds the numbers to facilitate mental calculations. In this

 

 

 

 

 

 

scheme, the student may have rounded 36 to 40 and worked with the remainder four,
because it should be part of the answer. In this try he might have estimated, and the total
he found was 64. This showed that he was wrong in working with 20. This explains
why it didn’t matter whether he said 64 or 54, because both were more than 50;
consequently they were wrong anyway. In his second try, he may have decided to work
with 10, but he had the remainder four from 36 in mind, which was part of the answer,
so he said 14.

Another example of solving the problem of buying 50 cruzeiros of bread costing
4.50 each is from Mauro. He explained that with 10 cruzeiros he could buy two loaves

of bread and there would be one cruzeiro left. Fifty cruzeiros could buy 10 loaves of

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bread and five cruzeiros would be left. Five cruzeiros could be used to buy one more

loaf and he would still have .Fifty left. In this case, I could say:

10 cruzeiros to pay 2 loaves of bread 1 cruzeiro left
50 cruzeiros to pay 10 loaves of bread 5 cruzeiros left
5 cruzeiros to pay 1 loaf of bread 50 cruzeiros left

Making successive approximations

To solve the same problem described in the previous situation about buying candies
(Problem #2), two examples of successive approximations can be shown where students
performed successive and implicit additions when solving the problem:

Anita: I don’t know how much the milk costs.

Eduardo: 43.

Roberto: There’s seven lefL.. seven minus... How many candies can I buy?

Anita: four, no... each candy c05ts two, so I can buy... Let me see, one... two...

three. (She was counting in her fingers while talking.)

This could be expressed as follows:

1 candy 2 candy 3 candy

2L0 200+200 400+200

|__\.../.__\../

Anita was adding successively each amount she would pay for each candy she had
added.

A second case shows students solving the same problem:

Silvia: I can buy two candies because I’ll have five left.

Nelio: I think it’s right because two plus two is four and there’s 1 left.

Vera: There’s five left, milk costs 45... To complete 50 it would be five left. So

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the candy is two, plus two is four, there wasn’t one to buy three candies, it has to

be one left.

In the situation of buying loaves of bread costing 4.5 each (Problem #1), other
examples of doing successive approximations appeared. Students were looking for the
number that could be the dividend in the division of 50 by 4.5. But students did not
complete their reasoning to find the answer. In the following "example, it is important to

note that the student had in his hand five 10 cruzeiro bills.

I thought like this... But I’ll take out (showing a 10 bill) 50 cents: four plus four is
eight, with this amount I can get two loaves of bread, and plus this (another 10 bill)
I can buy four, this here buy six, and this (another 10 bill) buy eight, with this
(another 10 bill) I buy 10.

It could be argued that the situation of having five 10 bills suggested successive
additions to the student. Note also that the student uied to round the value of four—and-
a-half to four to facilitate the calculations. Two other students did the same rounding.
One of them, Laura, explained her reasoning: "I thought: 10 to two, remains one.

Then, I put the same value for 10 loaves of bread and five remained. With five I can get
one more loaf of bread."

Other solutions were described where students made successive approximations by
adding to find the solution to the problems, such as:

2 + 2 + 2 + 2 + 2 = 10 loaves of bread. Or, in problem #2 about buying candies:

1.50 + 1.50 = 3.00 2 candies

 

 

adding 1.50 3 candies

 

adding 1.50 4 candies
3.00 plus 3.00 is equal 6.00
6.00 plus 3.00 is equal 9.00
Then 10.00 buys 6 candies and 1.00 remains.
One very interesting resource for solving problems in informal situations was the

use of drawings. Laura drew the situation described in problem #7 on paper, and, by

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making successive approximations subtracting, she got the solution:

1.5
1.5

1.5
1.5

6 sheets

And explained what she thought: "I spent one here, one here, here and here
(showing each one of the shelves). Four are gone. Two sheets are remainders. Then I
cut them in halves."

Roberto had done the same. They both took out what was clear for them - one
sheet for each shelf. The ratio of two sheets for four shelves became easier to perceive.
They could easily take half out for each shelf again. Other students added one-and—a-
half four times and checked to see if the total was six.

Edna spent less than 10 seconds to give the answer for the problem. She
explained:

I thought that if I spent two sheets of paper on each shelf, it wouldn’t be enough

because two sheets for one shelf, two sheets for another shelf, would be four

sheets, adding two would be six sheets, so it wouldn’t be enough, then it will be
one-and—one-half.
When the interviewer asked her why it would be okay being one-and-one half, she said:

"Because one-and-one—half one-and-one-half I get three sheets of paper. Adding

one more (shelf) four-and-one-half and one (shelf) more it will be six (sheets)."

Using the remainder to balance the division.
When the problem presented was a non-exact division, the students might or might

not leave the remainder in either of the situations. When they were making distributions

119

of loaves of bread among two classes of students, or organizing two teams with seven
students, they realized that in both situations they would have a remainder. Some of the
students answered leaving the remainder untouched, and others preferred not to. But the
aspect that seems to make the difference in the decision whether to omit the remainder or
not is the context of the problem. This can be shown in the following problems:

1) seven children want to play queimoda. ( a Brazilian game). How could you divide
the two teams?

2) There are 45 loaves of bread to be divided among two classes of first grade students.
How could this division be done?

Both of these problems have the same mathematical structure: There is a set to be
divided into two subsets, the first one being represented by an odd number. In the first
problem, the amount refers to persons, which cannot be divided in a continuum.
Supposedly, this would require a division with a remainder.

In the second problem, although the amount is represented by an odd number, it
refers to a loaf of bread, which could actually be divided. In the answer to this kind of
problem, it is not necessarily required that the student leave the remainder out. The
Student could divide the one loaf of bread left into two pieces and have half for each
class. However, the students in this study showed a different attitude related to these two
problems. There was no apparent preference in continuing the division or leaving the
remainder out in the second problem. They suggested b0th ways. But, in the one about
the division of the two teams, the remainder was extensively discussed. They tried to
find different solutions which would avoid the remainder, and at the same time, continue
being fair in creating the teams. They did this by describing the same strategies they use
in real situations when they are playing. The most common strategy was defining one
extra life for the team with less players. In this case, the answer to the problem was to
have one team with four children and the other team with three childrens and one extra

life. This meant that one of the children could have the role of two childrens in the

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game.

Another strategy was to balance the two teams, not in terms of the number of
players, but in terms of the strength of the two teams. It is important to say that the
strength of the children in this game is very important. This is because the best player is
the one that can throw the ball with enough strength for it not to be caught by the enemy.
Other strategies were: 1) one player would play on one side for half of the game and on
the other side for the second half of the game, and 2) the remaining child would assume
a different role in the game such as being the judge or catching the ball when it was out
of the field.

The common point in the discussions about the remainder was to avoid having the
child out of the game. The following conversation in a group of children illustrates
several aspects discussed in this regard, particularly the desire of not having one out:

Fernanda: You cannot because it is necessary to have eight... only if one stays out

to be chosen when someone loses.

Laura: Only if it is four players on one side and three on the other side....

Paulo: and one keeps one life.

Interv: What is another possibility?

Paulo: One stays out.

Fernanda: And if he does not accept this? He has to play if not he would mess the

game up....

Paulo: Or then, the team which has two girls would have one more player on its

side.

The same reaction related to the remainder occurred when they were dealing with
two oranges to share with three friends. This problem presents the same mathematical
structure that the others discussed before, as far as the idea of sharing is concerned.
However, this problem presents one important, distinctive aspect. The set to be divided

is smaller than the number of people with whom the oranges should be shared. The

121

children’s preference was to divide the two oranges in two pieces each and give
one piece to each friend. The remaining one- half would either be divided again
in three small pieces, or it would be given to a fourth person defined by the
children themselves. This could be the younger brother, the grandmother or the
teacher, or the children themselves. One child’s strategy to avoid the remainder
was to divide each orange in three pieces and give two pieces for each person.
The following conversation among three students illustrates this:

Maura: I know. Cut the two oranges in three pieces each and give two

pieces for each friend.

William: Then there will be a remainder of one

Maura: No

William: Yes, and you take the other part and divide again.

Maura: No. There are two pieces for each. You cut the two oranges in

three pieces each and give two pieces for each friend.

Laura: Cut in two, and you have four. Then there will remain one and will

be one for each.

William: Cut the two oranges. Give one piece for me, another for M., and

another for....

Laura: The one remainder, cut it in three. So it will be one part and a half

for each one.

From the cases and situations I have presented here, it is possible to see how
much children can do in terms of mathematical reasoning in the everyday
situations they face. This contrasts strongly with the emphasis on the application
of algorithms when students are solving the same kind of problems in
mathematics classes.

In the final chapter that follows, I will discuss the possibility that there
exists two different ways of doing mathematics depending upon the context in

which children find themselves.

CHAPTER VI

CONCLUSIONS

Discussions with students about mathematics contrasted with my observations of
them in situations where they used mathematics. Thw difference between our discussions
and my observations reveals the distance between what students understand through
mathematics and their experiences outside of school.

One of the key differences I will focus on is that students apply algorithms to
school problems mechanically, without thinking about why, or whether, the resulting
answers are reasonable, while they don’t do this in “real world” mathematical situations.
I will approach these differences or contrasts taking into account how: 1) the
characteristics of students doing mathematics in school and out of school were shown to
me, and 2) students performed in both situations. I will explore how these and contrasts
reveal the possible existence of two different views of mathematical knowledge. To
conclude, I will reflect on the roles of particular aspects of the students’ social and
cultural context in the construction of their notions of division.

The first sign that I might be dealing with two different things when approaching
students’experiences with mathematics inside and outside school occurred in a
conversation I had with Fernanda. She expressed a clear distinction between her own
idea of division and what the teacher expected from her when doing division at school.
In fact, she pointed out the two aspects that I would look at in this study the: 1) student’s
idea of division and 2) the division taught in school. Here is that segment of our

conversation:

122

123
Fernanda: I thought that division was like this: 10 divided by five was five and

five. At home, if I change 10 cruzeiros, I split five and five.

Interv: Is it a division?

Fernanda: Ah, I think it is, but when I have to do it in the notebook, it is not. IfI

do that, the teacher will consider it wrong.
As I was talking with Fernanda, I wondered why she thought that her way of doing
division would not be division in the classroom context. By expressing her idea of
division in two different ways, at home and at school, she suggested to me that there
were two different views: 1) one coming from how she does things in her daily life, and
2) the other from her understanding of division from classroom teaching. Thus, I should
distinguish “real world” from school mathematics. This might clarify the difference
between how students perceive the nature of mathematical knowledge in the real world
and in school.

As discussed earlier, the real world to students I worked with is a place where
limits are defined according to different, concrete positions people have in different
places. Decisions are made according to interests and the particular conditions of power
of the decision maker, as well the limits of the situation where the decision has to be
made. Consequently, there are always different sides to be considered in a decision-
making process. The decision varies according to who is deciding and how it is made.
The power of doing things--deciding, taking, or giving--is related to the ideas of
authority that students recognize, and to whom they submit. Power is also related to good
social and economic conditions, in contrast to poverty and inequality. According to these
differences perceived by the children, there is a logic to what is possible for them and
what is not.

This logic of the possible is defined in terms of what they realistically have
access to or believe is attainable. This can be seen at different levels. For example, they

know it is possible to vote in a presidential election, but it is not possible to interfere in

124
the president’s decisions about what has to be done. It is possible for them to have a job,
but not to decide about a fair salary. It is possible for them to have access to school
education, but it is not possible for them to go to private school because they cannot pay
for it. In sum, as Freire has pointed out earlier in his work with illiterate adults in Brazil
(1970) the idea of what is possible is related to the power they have and their conditions
of existence.

In contrast, school life represents a break in the students’ daily experience
outside school (Singer, 1988). This does not mean that what is going on outside school
does not affect school life have an influence on it. The image that better represents what
I observed in the school site is a filter. This means that students are taken from the limits
of the real world and put in a place that is presented as different. This difference is
defined as the school's social distance from real world events, and by the student's
treatment in the school’s context. The case of meningitis is an example. Meaning it is
spread, and several cases had been reported in the school's neighborhood. While the
entire community knew of the existance of the disease and, while insufficient food and
housing and inadequate arrangements for controlling the disease were contributing
factors, the school appraoched the situation as if it resulted from inadequate personal
hygiene. To the students, the school’s approach made it appear as if washing their cups
and refraining from kissing one another would stop the disease.

In the same way, mathematics teachers presented numbers to children as being a
world unto themselves. That is, numbers alone, without context, were assumed to have
meaning. Therefore, students were being taught the mechanical manipulation of
numbers without realizing that mathematics actually comes from life outside school and
that their experience is full of mathematics. When students reported how they dealt with
situations requiring mathematical reasoning inside and outside of school, they spoke of a

different kind of mathematics.

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School mathematics follows an approach within the parameters of a formalist
tradition of mathematics (Machado, 1987). Students think of mathematics as a
manipulation of numbers according to certain rules. By dealing with numbers,
mathematics is abstract, value-free and exact. Numbers speak for themselves about truths
within a field of centrality where it is possible to say exactly what is right and what is
wrong. Thus, school mathematics operates within a fictional world, working according to
its own rules, and where its logic is different from the logic of everyday life.

The following conversation shows how different the approach is when school
mathematics is used. Three students and I were discussing some situations the students
face in stores in the school’s neighborhood. We had just talked about buying a liter of
milk with 50 cruzeiros and candy with the change they got. Each one of them answered
me, correctly, how much candy they could buy with the change. The figures given by
each child were different according to the price they paid for the milk. After this, I asked
them to solve the problem of how much candy I could buy with five cruzeiros at one-
and-a-half cruzeiros for each candy. Here is how our conversation proceeded:

Fernanda: It’s adding? What calculation could it be? Let me see how much this
will be.

Rita: It’s adding? (asking herself.)

Interv: Why are you adding?

Rita: Mine (the result) is already wrong. I will make every calculation: divide,
multiply,

subtract and add... but adding didn’t work.

Laura: I finished. The answer is 510.

Anita: But, buying 510 candies?!? (They laugh.)

Interv: It means that I go to the store with five cruzeiros and can buy 510 candies?!

(with surprise.)

Rita: GOSH!! Only if it is with “fish man” (All laugh again.)

126
I proposed that they should not think in terms of pencil and paper answers and

asked again, if I had five cruzeiros, aproximately how much candy could I buy. They all
answered “two” or “three.”

This situation shows some of the limitations the students encountered when they
tried to solve the problem using school mathematics. First of all, the students did not
approach the action of buying candy as a real situation. The evidence for this is the
absurd answer (510) Laura gave to the problem in contrast with the estimation (three
pieces of candy) she made at the end of the conversation. The first answer was the result
of the wrong division. The answer she gave was only a number resulting from the
calculations she had made. It was not a reasonable conclusion based on real events. To
reach a conclusion about the plausibility of the answer, the student would have to think as
if she had the five cruzeiros to buy the candy. She would then realize that it was only
possible to buy two or three piences of candy. Second, the emphasis was on the algorithm
to be applied, n0t on the action of paying one-and-a-half for a certain amount of candy,
which could not exceed five cruzeiros. The evidence for this is the fact that all three
students were looking for one possible algorithm to get the answer, it could have been
adding, subtracting, multiplying or dividing. Finally, Fernanda seems not to make any

sense of the algorithm she used to divide. She wrote on the paper:

1.50 '5

-1 510

E-
- 5

00

When I looked at what she had done in connection with my observations of her in

the mathematics classes, I observed that she did not understand the application of

127

the algorithm for division. One example of this is the numbers on the left side that

are only repetitions of the divisor or quotient.

Another situation occurred when Helena uied to invent something about
mathematics based upon picture # 7:

Helena: One lake adding three of these pipes would be four.

Inter: Can you add a lake and pipes?

Helena: In mathematics, it can be done.

From this conversation, I realized that in saying “in mathematics it can be done”
she was distinguishing two different meanings of “possible.” Possible here goes beyond
her own sense of possibility in the real world. Adding lakes and pipes was possible in a
fictional world with no real context and where explanation was based only in the
correctness of the algorithm applied.

In other words, what is possible in the real world is different from what is possible
in school mathematics. In school mathematics, a possible solution is the result of a
mechanistic application of an algorithm. When it does not work, the result is regarded as
impossible to obtain. When students are solving school problems, they apply the
algorithms they have learned at school and they are supposed to find the correct answer.
There is no need for reasoning in terms of plausibility, as is necessary in real situations.
The problem buying five cruzeiros worth of candies costing one-and-one-half each was
supposed to be real and meaningful because it was taken from a concrete situation
(buying candy) in students’ experiences outside of the school. However, students
approached it in the same mechanistic way described above. This is why Laura gave
more than 500 pieces of candy as a possible answer with the school mathematics
approach. When we discussed the same situation, there was no doubt in our conversation
that five cruzeiros would not buy more than a few pieces of candy. The school
mathematics which emphasizes algorithms and mechanistic working with problems does

not leave room to reason about absurd answers in the same terms that children would do

128

in daily life situations. The text of the problem, or the words that are used to describe it,

are ineffective in leading students to approach it as they would a real situation.
Decontextualization of school mathematics seems to play an important role in how

students reason and solve problems in school. Tables 1, 2 and 3 summarize how students

performed in both in and out of classroom situations when solving word problems.
Table 1

Results of Students Solving Problems in the Classroom Context

i # of Correct Wrong Answer No
L Problem M 7m“ __ -W__________n Work k;

 
  

       

 

 

 

#1 27 4 20 3 2
#2 27 19 7 1 4
#3 27 19 8 - 5
#4 27 4 l9 4 3
#5 27 8 14 5 2
#6 27 8 l4 5 2
#7 27 6 18 3 4
#8 27 7 16 4 5

129
Table 2
Results of Students Solving Problems in the Real World Context

5 ' ‘ _ __.___.________________ ‘ ose ‘7.
L Procblm S‘U‘SMC“_ _______A__ Jdma‘ir

     

 

#1 26 8 4 1 13
#2 26 25 l - -
#3 22 22 - - -

 

 

 

Table 3

Results of Students Solving Problems

lassroom ontext Real World ontext l;
# of Correct Solution # of Correct Solution H
Problems Students F % Students F % i

 

l 27 4 14.8 26 8 30.8
2 27 19 70.4 26 25 96.1
3 27 19 70.4 22 22 100.0
4 27 4 14.8 26 22 84.6
5 27 8 29.6 23 12 52.1
6 27 8 29.6 19 12 63.1
7 27 6 22.2 4 4 100.0
8 27 7 25.9 18 11 61.1

 

 

' 130

An example of my previous point is shown in problem #6 in both tables. This
problem is about 45 loaves of bread being distributed between two classes of first grade
students. In the classroom context, only eight out of 27 students (29.6%) solved this
problem correctly. In addition, two of them gave only an answer, but did not show the
work they did to arrive at it, and five students (18.5%) did not even try to solve the
problem.

On the other hand, 12 out of 19 students (63.1%) were able to respond correctly to
the same problem outside of the school context. In addition, three out of seven who
could not solve it, gave reasonable approximations of the right answer. Thus, the
likelihood of absurd answers diminished when students were solving problems in the
interviews. As shown in the tables, when the problems were solved in the classroom
context, the majority of the students either did not try to solve the problem, solved it
incorrectly, or gave the answer without showing their work on the problem.

Table three summarized part of data in table one and two combining percentages
of correct answers both in classroom and real world contexts. Table three shows
evidence that children's level of correct answers is hiwer in the real world context as
compared with classroom context.

AnOther important aspect to observe is the richness of the students' reasoning when
they solved the problems outside the classroom. In problem #3, for instance, the
discussion about the remainder when dividing seven kids into two teams did not occur in
the classroom situation, though it did in the interview. Also, as I mentioned in Chapter 4,
the children produced different strategies for solving the problem, as they considered
various aspects involved in a balanced organization of the two teams. This became
evident in both the discussion of the team's definition during the interview, and when
they were playing on the street.

In the classroom. when students were solving problem #4 about three friends

dividing two oranges. none of the students performed the algorithm correctly (dividing

131
two by three), although six students had described this as the operation to be done. One
student used a drawing to find the answer. In the classroom, only three possibile correct
variations. Here is a case where different conditions and interests account for different
ways of dividing two oranges among three friends. When facing the same problem in a
classroom situation, students divided two by three, not two oranges among three friends.
even though the words used in the problem were the same.

In terms of the strategies students used both inside and outside of the classroom
context, the difference is far greater than I expected to find. In classroom situations, the
students looked for one of the algorithms taught, and applied it to find the answer. They
took the number resulting from the algorithm, whatever it was, and mechanically wrote
it as the answer, without reflecting on the result at least to see whether it was reasonable.
The teacher had told them to do this. She also has implied that if they performed the
operation correctly, they would obtain the right answer. Therefore the students had faith
in the teacher. Message and obeyed her command. Students did not use any strategies
to facilitate calculations, nor did they take any initiative to bring their experiences from
everyday life into the situation.

In contrast. when students were solving the same problems outside the classroom,
they used a variety of strategies to reason their way through these situations. Rounding,
making estimations, and simplifying were the most common ones that I found. Carraher
et al. (1984) illustrate particularly the Strategy of rounding. They argue that it is easier
for children to deal with tens, to facilitate calculations. Almost all studies about children
doing division outside the school context (Boero, 1989; Burton, 1983, Carraher, 1987
Kouba, 1989; Moser, 1982) mentioned. Other strategies I, tooo, had found among
students, such as making successive additions and subtractions.

In my study, the children presented sophisticated processes of mentally calculating
the answer. For example, some of the students reasoned in terms of proportion, made

balances of different amounts. and used some propertiesl of addition and subtraction.

132

Weiland (1985) mentioned suategies involving the distributive property of operations.
Schlieman and Magalhaes (1990) studied proportional reasoning used in shopping,
kitchens, laboratories and at school. All these strategies presented in these and others
studies show the variety of ways students solve problems. On the other hand, the
school’s mathematics leads to students’reasoning based on fixed algorithms. When
students do not understand the algorithms. the algorithms make it more difficult rather
than easier to solve problems. When this happens. learning mathematics become
mechanical, boring and lacks the rich capabilities already possessed by the students.

As I reflect on what the students said about mathematics and how they used it
inside and outside school, I am convinced taht we as mathematics educators have to
review our practice, if we want: 1) students to learn with understanding and 2) to

enhance the development of students' creative minds.

Consmggg'ng the Ngg’gn of Division

In this chapter I have been analyzing the contrasts and contradictions I found
between students' perceptions of school mathematics and mathematics in their daily
experience. As I mentioned, the three main sources for my premises have been: 1) social
and cultural context, 2) mathematical knowledge, and 3) mathematical reasoning in and
out of the classroom situation. Now, I would like to resume the initial frame work of this
Study and look at the basic points which have guided the study. The focus is on the
construction of mathematical knowledge as a process in which the learner and his/her
social interactions are central elements (Confrey, 1987). In this sense, the initial question
of this work is how do individual student encounter mathematical knowledge in school?
More specifically, what do students experience as they encounter mathematical
knowledge in construction the idea of division?

I have already said that this encounter could not be seen when I examined

information from school only. When I was in the classroom, it was impossible for me to

133

connect what students were learning about division in their mathematics class with their
experiences outside school. Therefore, it was necessary to become familiar with both
contexts, and then to look for connections. This procedure contrasted both contexts and
unveiled the encounter I was looking for. It became apparent in how students talked
about division, and particularly, in how they did division. This report is replete with
examples of how they did division in a variety of situations both in and out of school.
In their journals, I asked what division meant to them and they responded:

-"It is a very important calculation."

-"It is the opposite of multiplication. It’s to split numbers: it’s the division of

numbers."

-"It is to split, to distribute."

-"It is distribute, to share something."

-"It is the same as multiplying because four times five is equal to 20 and to 20

divided by four is equal to five. Then, if someone does not know multiplication,

the person also does not know about division."

-"It is an operation used by people."

-"It is sharing one thing with other people."

-"It is to split, to allot, distribute."

-"It is a number divided by another one."

-"It is when I have a biscuit and share with an0ther boy who is with me."

-"It is sharing by two persons or more."

-"It is 10 divided by two is equal to five."

-"It is sharing with others."

Taking a careful look at these answers, I observed that, although some students had

drawn from definitions they had heard in classroom situations, the majority of them
expressed the idea of sharing, splitting, or both. Sharing was the most frequent idea of

division students dealt with while I was studying them outside the classroom.

134

Sometimes they interpreted incorrectly the teacher’s explanation. This was the case
of Sandra, who answered that division is “the same thing as multiplication because four
times five is equal to 20 and 20 divided by four is equal to five.” In fact the teacher’s
explanation was that 20 divided by four is five because five times four is equal to 20. She
was relating division and multiplication as a process of taking equal amounts apart
(division) or putting them together (multiplication).

It is interesting to note that the guidelines teachers have to work with when
teaching division are based on two main ideas: Partition and measurement (Amaral and
Castilho, 1990). Sharing is seldom mentioned by authors when dealing with types of
division, although Zweng (1964) considered sharing as one distinct aspect of partitive
division.

I would like to call the reader's attention to several important aspects of division I
identified in how students talk about and do division. First, students distinguished the
division they did in numerical terms from division they did in broader terms, such as
sharing or distributing. Division in numerical terms was the one in which they dealt with
numbers, and where the emphasis was on numerical relationships or on quantities. This
was the case when students were dividing numbers outside of a “real context,” such as
numerical problems in the classroom. In this situation they might have had 15 divided by
three, and made calculations according to the school algorithm to find an answer. This
kind of division started and ended strictly with manipulation of numbers, and the correct
numerical result was what mattered. There were other times in the classroom when the
problem contained a had a depiction of some situation from daily life outside school, but
the division was guided by formal mathematics as it was presented in the school context.
In this case the problem referred to elements of daily life outside of school, but the
students did not assume this was the real situation. The following word problem used by
the teacher, in an attempt to be more connected with the students’ experiences is an

example: "Jose has 39 coffee packets to arrange on three shelves. How many packets

135
does he need to put on each shelf?"
This kind of word problem talks about things students know well, because they deal with
vendors at their stores frequently. In addition, this is the way vendors show the products
they have to sell, exposing them on shelves. Also, coffee is one of the most common
product sold, besides milk and bread.

When students saw this problem in the classroom. they had in mind only the
numbers involved and the algorithm they used to find out the right answer. In the real
situation, several aspects would be taken into consideration besides having an equal
amount on each shelf. In a real situation, Mr. Jose would be considering the size of the
shelf, its location, how much weight the shelf could support, whether he wanted to put all
coffee packets on the same or on different shelves, and so on. Therefore, when the
students have to solve problems within the classroom, they do not solve them as they
would in their daily experiences. This point is extensively demonstrated in Chapter 4.
When students think in numerical terms, the division of four by five, for example, is
viewed as impossible to do because the students say that they “cannot take five from
four. If it was four divided by four, it would be possible." This alleged “impossibility” is
because the students simply have not learned the algorithm for this kind of operation yet.
They were reasoning only in numerical terms, without considering the process of
division as a real possibility, symbolized in numerals.

On the other hand, students did divide two by three when they solved the problem
about the oranges in the interview. They found several ways to divide two oranges
among three children. In this case, they were not working in numerical terms, dealing
only with numbers. Students were dividing two oranges among three children as they
would be doing in a real situation. What is missing in the example described above is
the algorithm to put this operation in terms of school mathematics. In this sense, I have
asked myself whether problem solving is taught in schools through a process of

“domesticating” minds, (Freire, 1987) rather than stimulating creativity and freedom of

136
thought to find possible and intelligent solutions. In other words, do we teach
mathematics in our schools in a way that puts constraints on children’s minds, instead of
promoting the development of children’s intelligent reasoning? The evidence that leads
me to this question comes from my observations of how the children solved problems
outside school, compared with how they did in mathematics class. These students are far
beyond where the school is. School is not teaching problem-solving strategies. Nor is
school building on, acknowledging or even uncovering the intuitive, informal, but
sophisticated and complex strategies students have constructed and use outside of school.

Coming back to the discussion of the students’ ideas of division, the second idea I
found in their definitions is exactly what I was talking about earlier. Students did
division outside of school as an action of sharing, giving, or making a distribution. This
means that Students solved problems not in numerical terms, but as if they were actively
dividing something. The action was of sharing, distributing or giving. The emphasis here
was on the process of doing division, on the act of dividing. The correct numerical result
was not that important; the “how to” came first. This idea of division was so strong
among the children I studied that almost every time they talked about division, they
referred to situations where they were sharing, distributing or giving.

To corroborate this interpretation, I should mention that when I asked students
about things they could divide, they also answered by naming things that are part of their
own life, such as personal objects, toys, candies, food, and so on. The following
student’s statement is an example: “I can divide my things. I cannot divide other
people’s things.”

The most striking characteristic of this idea of division as an act of sharing, giving,
or disuibuting is fairness. Being fair, in the sense that is used in this meaning of division,
does not necessarily mean equal division. In the division of children into two teams to
play a game, for example, several solutions were proposed for the purpose of being fair,

and most of them did not just take into account the number of players on each team. In

137
fact, even when they had the same number of players on each side, other factors were
considered, such as strength and weakness of each player.

In the situation of sharing, as for example having two oranges to be divided among
three children, they showed clearly that being fair might not mean equal division. One
student gave me the following solution to this problem: “The youngster gets one whole
orange and the other two divide the second orange in the middle.” Or in another
example from S.: “I will keep one for me, give another to her (a friend) and she (another
girl) will get none.” This example again raises the issue of people’s interests when
dividing. The use of different criteria to make divisions in numerical terms or as an
action of sharing, distributing, or giving is never approached in mathematics class. In the
classroom division are made into equal parts is taken for granted in all situations. The
use of algorithms to make calculation is taught as the only aspect of problem solving
that is important in finding the right solution. The relative importance of the algorithm is
not discussed, nor its intelligent application to help with the solution of the problem.
Once more, I should say that the school and mathematics class, as they exist today, do
not offer the opportunity to discuss intelligently this kind of issue and to enrich the
students’ experience.

The third idea of division I identified is division in a physical sense. This means
division that implies splitting into two or more pieces, or groups. The physical aspect of
pulling apart, or separating is prevalent. In this case it is meaningless to talk about
fairness as in sharing, or in equal division as in numerical division. The point here is, the
results of this kind of division are not necessarily equal parts or groups with the same
amount. This is how this idea of division differs from the division in the classroom. In
splitting, or dividing in a physical sense, it does not matter if an equal or fair division
results. Students did mention this kind of division on different occasions when they
described people in the church on both sides of the saint’s image, or different directions

of the reflection of light (picture #4). In these cases, the results were not the same

138

amount on each side, nor was fairness the issue. The same comments came up when
they identified the division of the glass door as separating the people in and outside of
the building (picture #9), and the division of men working from men looking (picture
#2). Once more, students were not referring to equal numbers as a result of the division.
They were pointing out differences they could see in the pictures. when taken from two
different sides. Another example of this kind of division was when students divided
numbers as they were dealing with two pieces of one set. This was the case when
students were trying to solve the problems about figurinhas (problem #5), or about
buying candy with the change frm 50 cruzeiros (problem #2). In those cases, students
defined two sides or groups and tried to work out b0th to get the whole set. When
students approached these problems in this perspective, they dealt with two groups, not
necessarily equals, to make the whole set again.

In concluding, I would like to bring into this discussion some connections I found
in reflecting upon these ideas of division, and the frame I constructed earlier of some of

the students’ views of their cultural and social context.

Rfl tin n h uen ’I asofDiviin

The first connection I perceive here is between the idea of division as separation of
parts, and the strong distinction Students make between their own social condition and
the condition of those who have the power to decide and do things to improve their
lives. The polarization of rich and poor, those who work and those who pay, those who

ask for and those who give, might generate in some way a distinction between “us” and

’9 H ’9 ‘6

“them, ours” and “theirs, we” and “they,” and so on. The inequality the students
perceive in their social context splits or sets apart two sides in the society where they
live. In this way of understanding division, the requirement of equal parts has no
meaning. In fact, it goes against their own perception of their reality, where equal

division is not the rule.

139

In addition, fairness and equality are not requirements for division of this kind.
This might have to do with the students’ need to face unequal conditions of living that
they are in, considering Others they know and see. This condition of inequality opens the
door to the discussion of equal and unequal division. Looking at things this way, I began
to understand why students talked about being fair when dividing instead of being equal.
Fairness is a requirement of their way of dividing, while equality is part of a fictional
world which they do not have access to fully understand. This fictional world could be
compared to life within the school. It is real because it is part of their lives, but it is also
fiction as long as students cannot understand it.

The immediate implication of this is having students dealing with two different
meanings of division at the same time, and having to make sense out of it. Students
might think about division inschool and in reality. In their social context, there is fairness
and unfairness. and both are embedded in inequality. In the school context, the idea that
is conveyed is of equality and equal, and both are embedded in inequality. These
contradictions, between equal and equality embedded in inequality, might reinforce in
students the approach to division they have in the school context. This explains why
students do so differently when dividing in and out of school contexts. And, this could
also be one possible explanation for the distance from which they approach mathematics
themselves. The filtration process the school uses to approach mathematical knowledge
may suggest to students a fictional world of mathematics where, again, they do not have
access and they do not need to fully understand, but they do have to know and follow the
rules of functioning.

The apparent passive acceptance of rules of functioning and conditions of
inequality in the context of this study might be explained, at least partially, by the
students’ perspectives of power relationships. Students manifested their vision of power

as being embodied in the figure of someone who can decide, do and change things.

140
However, they see themselves as, powerless. Perhaps this condition of the
impossibility of doing things might explain why thethe students accept and refer
to different ways of dividing: l) the division they do, 2) the division others do,
and 3) the division that is done within the school.

The second connection I see is the idea of division as an act of sharing,
distributing and giving with the students’ feelings of being part of a group. It
seems to me that they saw themselves as participants identified with one side of a
society that they perceived as divided. Their side is the one where people share
and support one another, dividing things they have among each other when it is
required or necessary. The action of dividing as sharing, distributing or giving
away could be related to a feeling of common identity developed among those
who are powerless. The mutual action of dividing could be understood as an act
of self-protection and survival in a hostile environment of inequalities.

Finally, I see a third connection in the numerical idea of division and its
identification by the children with the other side of a society perceived as divided.
The connection is drawn from the parallel between two sides: 1) the artificiality
and abstractness of school mathematics and 2) the distance of the sphere of
decision, power, and control from their reality. Due to its artificiality and
abstractness, the numerical idea of division relates to these children’s inability to
solve everyday problems at the level they could if they had the power that
knowledge confers on them. In doing so, school mathematics does not serve
students in their need to understand, explain and hopefully transform a reality
they perceive as unequal. This sphere is outside of students’ understanding, and
thus their ability to explain. The teaching of mathematics based on these ideas
reinforces and justifies their position and condition of inequality. Students take
this inequality for granted within a fairy tale world and fantasy. Their apparent

conformity can be explained once more as being a mechanism of survival and

141
self-defense. The equal division in numerical terms could be related to equal division

where things are not real.

To conclude, I must say that these reflections of mine on the construction of the
ideas of division are n0t final. My intent was to shed some light on the learning of the
concept of division. I think that cultural and social connections could help teachers
understand and deal with learning situations in mathematics classes. I also must say that I
believe mathematics should be a channel to improve students’ comprehension of life, and

“an instrument in their action as subject,” Freire’s (1970).

Implications for Praeo'oe and Recommendations for Fonher Reseageh

Findings in this research suggest that school mathematics as we teach it now in
Brazilian schools should be reviewed in three main ways: 1) curriculum organization, 2)
content knowledge we choose to teach, and 3) the way we teach it.

In the school where this research was conducted is similar to a great number of
Brazilian schools. It takes the format of a set of disciplines, which in their turn, form
blocks of content knowledge similar to bricks that can be used by children to organize
the experience they might have at school. Therefore, teachers at schools seem to assume
that children will build up their own knowledge as if they would be building up a wall of
bricks.own knowledge as if they would be building up a wall of bricks.

The research done has shown that children use a variety of different strategies,
always based upon their day by day experiential world, when doing their mathematical
reasoning. This line of research has a main implication for those in charge of reviewing
curriculum organization, especially in mathematics education.

My research suggests that mathematics as a science does not result from physical
impressions of quantitative relations existing in the world within a child’s mind. Also, it
should not be taken as a one single person’s lonely construction. Rather, knowledge
gains its meaning from shared social interpretation which is its use (Lerran989). In

this sense time and space, as well as social content and culture, become particularly

142
important in dealing with curricular organization. Being more specific, formal
mathematics cannot lose sight of existing relationship between mathematical knowledge
and children's comprehension of their existential space. The world at school, that is
expressed, among other things, through the knowledge it imparts, should not be opposed
to the world and experiential knowledge children have within their familiar, social and
cultural life. My research has shown deep ruptures--between the world of school and the
world of life--that review of the curriculum, such as I have suggested, would prevent
from widening. Much of the needed review, of curriculum organization can be
summarized as adopting a new vision of the role of schooling in the life of individuals
and the society.

In addition to the review of curriculum organization, content to be taught also
needs review. One should consider first that adopting a broader vision about the role of
schooling, as indicated before, also requires a different perspective on the choice of what
should be taught in mathematics and on the way it should be taught.

A commonly used method to teach mathematics in Brazilian schools today is to
emphasize the use of memory, calculations and repetitions. This way of teaching is
clearly connected to a vision of schooling where mathematics teaching is taken as
transmitting mechanical processes of calculations, involving abstracts formulas that once
put together are assumed to be the content knowledge of mathematics. Findings in my
research suggest that there is a need to move toward a new direction, in school’s
mathematics where students actively participate in the construction of their
understanding of mathematical knowledge. In addition, my findings also indicate that
students' constructions of their understanding frequently takes place through
interpersonal negotiation of meanings during their day-by-day experiences. An
implication of these findings is the need for schools to find ways to integrate into the
students' school life their sense making, achieved during those interpersonal negotiations

happening in their daily life.

143

Accepting that as true has another implication for the way mathematics should be
taught. A class of mathematics where construction of meanings by interpersonal
negotiation is valued should be rather different from the current ones. The mathematics
class should be transformed into a locus for collaborative and integrative learning and
teaching. It should take into account the social and cultural context in and out of school
as the ground for teaching and learning. Based on that, the students' activities of
intellectual inquiry should be emphasized. They should incorporate discussions and
collective dialogues exploring mathematical ideas as routine procedure in the classroom.

Another important aspect in mathematics classes that should be reviewed is how
teachers teach computation. My research suggests that children would do it better,
should the mathematical algorithm used to make the computations be the result of the
student’s effort to make explicit his own way of reasoning. The implication of these
findings is that teachers should take advantage of the different strategies students use to
make calculations and help them to build a bridge toward the formal algorithm. In
addition, teachers could explore these different strategies to enlarge the student’s
understanding of mathematical principles and properties related to those strategies. For
example, when students use successive additions to make divisions, teachers should
relate these procedures with the associative property of addition. In another situation,
students round numbers to facilitate calculations. It could be related with rounding
numbers in solving problems where estimation of possible results are used when dealing
with more complex situation problems. There are a variety of mathematical properties
that teachers find it difficult to teach to children due to its abstractness. However,
teaching procedures would be more effective if the strategies students use out of school
could be used. They could be taken as linking points between the student’s
understanding of mathematical situations they deal with in their day-by-day life and
meaningful mathematical properties they could systematize in their classes of

mathematics. Summing up, student’s mathematical reasoning in their daily experiences

144
out of school should be brought into the classroom situation, and be used as a reach
source and resource to enrich the teacher’s formal mathematics teaching strategies.
Finally, as implied this study, the following questions surfaced that should be
more deeply explored in further research because such research would contribute to new
developments in mathematics education.
1) How can students bring into the classroom situation the same autonomy they
have in dealing with mathematics out of school?
2) How can students deal with mathematical situations out of school after
completing their schooling process?
3) What can mathematics educators do to make school mathematics more effective
in contributing to students’ computation competency out of school?
4) Will the use of more complex strategies to solve mathematical problems at an
early age improve the child's mind?
5) How can school mathematics empower the student to understand their social and

cultural milieu?

APPENDICES

145
APPENDIX A

List of the scenes in the pictures used in the interviews:

1- A car passing a sign indicating 60km/h speed limit.

2- Several men from a paving company repairing potholes in the street.

3- Several musical instruments and pictures of two men talking.

4- A saint’s image being carried by four men, and a crowd gathering in the church
surrounding the image.

5- A parking lot behind a building under construction where several men are working.
6- An ex-president, running for senator of one of the states located in the north region of
Brazil, leaving his campaign committee.

7- A water reservoir with three pipes in it.

8- A building with the world greve (strike) painted on the glass door.

9- A candidate runin g for Governor of the State of Minas Gerais being interviewed.

10- Brazil’s current President in a meeting at the presidential palace.

APPENDIX B

List of the problems presented in the interviews:

1- If you have 50 cruzeiros. how many loaves of bread could you buy if each loaf costs
4.50?

2- If you spend 45 cruzeiros to buy milk and pay with a 50 cruzeiro bill, how many
candies could you buy with the change?

3- Seven kids want to play queimada. How could you divide the two teams?

4-If you have two oranges to be divided with three friends, how much orange could each one
get?

5- To organize a collection of 150 figurinhas in two albuns, how many figurinhas should
I put in each album?

6- There are 45 loaves of bread to be divided between two classes of first grade students. How
can the canteen woman do this division?

7- If you use six sheets of paper to cover four shelves. how many sheets of paper will be
necessary to cover just one shelf?

8- Ms. Luiza has 30 cups of milk to be divided among 30 students in the third grade, but
only 20 students came to school today. How can Ms. Luiza divide the cups of milk

equally among the students? How much will each student get?

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