Maria do Carmo Santos Domite

The aim of this study was to highlight the dual relationship between dia­logue and action. To address this aim, I examined problematization and problem posing, which have the same kind of dual relationship. Problematization is a cognitive movement, not yet clearly delineated, made up of a push and pull between the asking of questions and the finding of answers about what is hoped will evolve into a problem. With the process of problematization as a starting point for posing a problem, this study had a double purpose: firstly, to show that asking questions is fundamental in order to learn; and secondly, to critically discuss the view that problematization starting in a real situation can be thought of as a proposal for a mathematics education in which life experience is used to develop students’ mathematical knowledge. Some evidence is presented of students’ attitudes in a sixth-grade mathematics class that employed problematization and problem posing in the process of teaching and learning. The author is a lecturer (research field: ethnomathematics) at the Faculty of Education of the University of São Paulo, Brazil.

Problem Posing and Problematization in Learning and Teaching Mathematics

In this paper I reflect on the role of posing questions for mathematical learning. In other words, I attempt to reveal that the act of posing problems, especially on the part of the students, is an effective means of achieving positive learning experiences. It is important to stress that a premise of this discussion is my belief that knowledge starts with a motivating question.

This premise is implicit in many of the theories of modern cognitive psychology and modern pedagogy, which are the psychology and pedagogy of action and interaction. In fact, over the last 50 years, through pragmatic studies and genetic psychology, research in the field of cognitive psychology has been directed by an epistemology of interaction. The messages this field has sent to the pedagogical field are essentially the following: the teacher should learn how to ask questions – instead of just explaining the lesson content – and how to lead the students to respond actively – directing their own reasoning – to obtain appropriate answers and ask their own questions. The principal message is that the individual is stimulated to think precisely when he or she enters into a state of doubt.

The field of pedagogy has been influenced by dialogical theory, in the sense that teaching must start with the student’s discourse. Freire (1986) rejects a non-dialogical attitude:

“This I call ‘castrating curiosity’. What is happening is a movement in one direction, from here to there, and that is all. There is no return, and there is not even a question; the educator, in general, gives the answer even if nothing has been asked him!”

It is essential to clarify that for my research a question is also a problem. For instance, a question becomes a real challenge for the learner when it demands an answer that is not obvious – in such a case, question and problem become one and the same.

More specifically, this study aimed at understanding the process of posing a problem – understanding the movement that occurs within pedagogical interactions towards the formulation of problems. This movement, not yet clearly delineated, is made up of a push and pull between the asking of questions and the finding of answers to what is hoped will evolve into a mathematical problem. This dynamic movement towards posing a problem is what I have called problematization.

If all goes well, problematization ends in a problem. This process need not end in a mathematical problem, but in a mathematics class, we expect that the search for a well formed problem will lead the students to achieve some level of mathematical learning. In effect, any learning linked to problematization begins when the problematization process is initiated. In this research, I was concerned with a special kind of problematization that begins with questions formulated between students and their teacher and generated from their social context.

In short, the aim of this research was to highlight the attitude that the students assume when the educator, in addition to taking seriously action and dialogue within his or her pedagogy, adopts an externalist approach to mathematics. In my analysis of the approach – an approach in which the teacher takes into account the socio-cultural context in the learning-teaching process – I have attempted to understand the meaning, worth, and role of this particular problematization process in mathematics classes from first to eighth grades.

Taking the process of problematization as a starting point for the learning and teaching of mathematics, this paper has a double purpose: first, to help educators accept the premise that question posing by the students is fundamental to learning; and second, to critically discuss some of the so-called active methods commonly adopted in mathematics education – for instance, problem solving methods in which the problems have been previously formulated by the teacher.

Knowledge of My Own Cognitive Processes

My study of problem posing was unquestionably motivated by a desire to appropriate a process of problematization in order to find effective arguments for advocating it as a means of attaining knowledge. My own problematization process was created as a result of my effort to sift through the confusing sets of questions and answers that emerged from my initial question: how can a teacher introduce and apply the process of problematization in the process of teaching and learning mathematics?

Such an effort and result may seem inevitable, but at least after a certain stage of my research experience, I became conscious of my own perplexed mental state and made a deliberate effort to delve into and learn from it. I felt that I always knew what I was talking about because I myself had formulated the questions. They were not handed to me by an external source, nor did I discover them by chance, they arose from my own perception of the relationship between the different factors to be taken into account.

I should also stress that the act of reflecting upon my actions so as to understand when and why my problematization processes, and my students’ as well, reached the stage of clear formulation, led me to understand that there is a close relationship between thinking and acting. One does not precede the other, however, even though metaphysicians might say action follows thought. Thought and action are dialectically related, and each serves as a source to motivate the other. In this way, we act, and we gain knowledge as a product of our action.

Another important issue is how my thoughts themselves developed in a dialectical manner. Many opposing ideas coexisted in my thinking during this research. For example, on the one hand, at times, I thought it might be anachronistic to use a problem posed by the group and generated from outside mathematics as a starting point for teaching and learning, given that mathematics in and of itself offers ready-made answers or formulas that I already knew how to teach in a interesting and dynamic way so as to capture my students’ imagination. On the other hand, my thinking led me at other times to assume a radically different position, following the words and ideas of Freire (1986) that “it is only through questions that one must look for answers” (p.46). In other words, if there are no questions, there will be no lessons. Indeed, from such a Freirean perspective, one could say that if there is no question, there will be nothing to say, nothing to teach, nothing to take into account.

Reflections Underlying the Problematization Process

Before discussing the work itself, I consider some answers to the following questions: a) why did I choose to investigate problematization as a means of learning and teaching mathematics? b) why did I opt for problematization and problem posing in the planning of daily classroom activities?

To begin with, I was deeply influenced by educationists and mathematicians who gave special value to active learning produced from the students’ social reality.

A second point is that although I became involved with problem solving as a means of learning mathematics, the necessity of having mathematics learning linked to a person’s life experience quickly became central. During my own process of transformation, I started rejecting ready-made problems – those all-too-well-known cute or ingenuous texts labeled as “problem solving exercises” – and, consequently, I started looking for questions that were generated within the student’s social reality as the source for constructing mathematical knowledge. Indeed, I started shifting the focus of the discussion of problem solving to that of posing problems, stressing and discussing problematization as the means of reaching the formulated problem. This pedagogic position came from my experience, observations, and reflection with respect to the following:

  • The problematization process is a productive movement towards social transformation, which means that the mathematics teacher’s attitudes could expand the students’ concept of their social life
  • The object of knowledge can be more or less internalized by the student according to his or her need or interest. This expectation is linked to the belief that mathematical knowledge begins with the question being generated by the learner: Indeed, the intention is “to place the children, their interest, their work and their experimentation in the center of the educational practice and to eliminate undesirable aspects of hidden curriculum” (Skovsmose, 1990, p. 116).
  • The pedagogical work produced by the situations that come from students’ social reality is a possible creative direction that motivates learning and teaching of mathematics. Others have shown significant effects and positive changes in mathematics classes that employ such methods – more specifically, when mathematics instruction is done through mathematical modeling.
  • As teachers of mathematics, we must be able to think/argue about the group’s production – in other words, we must be able to pay attention on the student’s process instead of just teaching a mathematical content. Lerman (1989) stresses the problematic of “content versus process” as one of the greatest obstacles for developing mathematical learning.

Third, the focus of the process of problematization is deeply related to my position regarding mathematics education, especially the type of problematization discussed above. The main components of this process include:

  • action
  • dialogue
  • an externalist approach to mathematics
  • the affirmation of the links between society and school/education and politics, and
  • the re-conceptualization of the notion of pre-requisites for learning mathematics

With regard to action, in my view the teacher’s role is to motivate the student to act towards new situations within his or her reality in order to investigate, analyze, and, if possible, to modify them. Action also involves the student’s own performance with objects and his or her own mental coordination. Indeed, action signifies the dynamics of the intellectual conflicts that students undergo when confronting the possibility of directing their own mathematical thinking.

By dialogue, I understand that the teacher assumes an attitude of patience towards different points of view and among these differences discovers sufficient similarities to establish a common target desired by the group. So, when I say action and dialogue, I mean at least two things: the first is acting on social situations or on material things with autonomy, on one’s own; the second, is doing and reflecting on things in social collaboration, in a group effort (from a group point of view). Communication and cooperation are essential factors in such a process of intellectual development.

According to D’Ambrósio, an externalist approach to mathematics is considered as: (a) a research field, in which mathematics establishes interactions between itself and other fields of study; that is, a mathematics at the service of the world; and; (b) a learning and teaching process, in which the teacher has to consider the social-cultural context in teaching the student; the student studies the situation of his or her social reality using mathematics as a language to understand, interpret, and possibly modify that reality. Therefore, contrary to an internalist approach that views mathematics as a classical explanatory form made up of a series of statements linked to logical connectives rooted in mathematics itself, an externalist approach to mathematics provides mathematical models and conceptual structures that are generated from phenomena external to the symbolic systems of the field. Roughly, an externalist point of view recognizes socio-cultural aspects as the principal factors of pedagogical action.

With regard to a close relationship between school and society, it is fundamental that teachers recognize education as a social process. Mathematics education must therefore have at its core the assumption that it is a social process. Promoting the link between school and society means at least two things: the first is that the problems of the entire school community should be discussed as an institutional and pedagogical problem; the second is that all efforts in teaching mathematical notions and techniques must be connected to their applications in daily life. In fact, the problem-posing pedagogy that I have developed, like that of Freire and D’Ambrósio, makes good use of Dewey’s vision of the necessity of teachers and students knowing and taking into account the life of the local community.

Finally, a re-conception of the notion of prerequisites for learning mathematics means that the process of problematization, by virtue of its very characteristics, demands a particular view of what are considered the necessary prerequisites for the attainment of new knowledge, especially when we are teaching Grades 1 to 8. This new conception opposes that employed in conventional mathematics teaching, which is the basic notion that mathematical learning takes a logical form in which a series of priori facts are necessary for new items to be learned. From the point of view of problematization, a prerequisite for mathematical learning is what and how the students know about the new mathematical fact to be taught and what kind of previous experience they have had with it in the realm of mathematics. In brief, a prerequisite in this new vision would be what the learner understands about the new mathematical concept rather than what the mathematician would like him or her to know. For instance, if a group of fourth graders are faced with a mathematical problem that requires comprehension of the concept of area and the students have never studied it in a systematic way, how does the teacher approach the learning situation? In this view, the teacher would certainly not impede the search for a solution to the problem or attempt to stop this search process to explain about area, step by step, through a mathematical model (e.g., present the notion of the square unit, the sum of such units, the multiplication and so on). Instead, the teacher should check the students’ understanding of the notion of area and, if possible develop in the students a more intuitive or concrete notion of it (e.g., invite the students to cover the region to be measured with sheets of newspaper to determine that the area is, say, 25 sheets).

Strategies for Problematization

To promote the process of problematization in the classroom, I suggested that the teacher initiate such a process by means of different strategies. These strategies had two main sources: one was my previous experience with the process of problematization and problem posing in mathematics classes, in which the students were involved in this approach to learning in the project “Interdisciplinarity via Generative Themes”. This project was developed within the special curriculum project “The Movement of Curriculum Reorientation” that was supported by the São Paulo Board of Education in 1989 – 1992. Both projects focused on Paulo Freire’s ideas that students’ natural practice and knowledge must be considered in the school process of teaching and learning. In 1989, Freire, the well-known Brazilian educator, became the secretary of the São Paulo Board of Education. My contribution to and participation in this movement concerned the field of mathematics and was centered on the teacher’s professional development. These strategies were also the focus of my doctoral research. Another source was the conceptions, views, and attitudes about learning and teaching expressed by the previously cited theoreticians.

The strategies I suggested that the teachers use to initiate and develop a problematization process were as follows:

  • To make evident a situation/dialogue from the school context: The teacher must consistently seek a situation that reveals itself as significant for the students and then help the students conduct a discussion about it so as to develop a problematization process. I call this the spontaneous strategy (SS).
  • To motivate/ask the students to choose a “generative theme”: The teacher invites the students to choose a situation from their social reality – in other words, a theme – and then must help them observe and investigate the facts underlying this theme so as to unleash a problematization. This is the generative theme strategy (GTS).
  • To present a theme to the students: The teacher chooses a theme to be discussed (preferably a theme that might help her or him to introduce some important mathematical content). The teacher’s art or action consists in provoking from the students questions linked to the theme. I have called this the provocative strategy (PS).
  • To analyze the solution to a solved problem: The teacher starts from a mathematical model that has already modeled a certain problem. The teacher can then present and analyze other problems, possibly arising in another context, that use the same mathematical instrument. I have called this the analogical strategy (AS).

The study took place during the first semester of mathematics in a sixth-grade class of a public school in São Paulo. The class had 36 students. They were regarded as the weakest sixth-grade class in the school, with a quite low achievement in mathematics and in other subjects. The average age of the students was 14 years. They were supposed to work in groups of four or five throughout the mathematics class period, and the composition of the groups was spontaneous.

Information was gathered by observation and the analysis of documents. The data reported in this paper came from the analysis of recorded dialogues – among the students and between them and the teacher – that might lead to a mathematical problem. The methodology for analyzing the results included a process that can be summarized as a synthesis of reflection and action. Indeed, it is an interventionist strategy that required the researcher not only to gather information but also to be the agent in the teaching situation in order to modify it progressively.

The next section contains part of two dialogues, each resulting from the use of one of the first two kinds of strategy noted above.

Results from the Dialogues

A problematization resulting from the use of the “spontaneous strategy”
Topic of discussion: “fertilizer recipe”
Group of three students – Mario, Paulo and Taciana – and the teacher

The context for the fertilizer recipe problematization was that when these sixth graders found out that a school custodian wanted to water the flowerpots in the school with water containing fertilizer, they became very interested in helping her to do it accurately. The teacher took the dialogue among the students into account and through a problematization process with further dialogue, they decided to devise a recipe for this kind of job.

Quantitative data: 7 equal-sized large pots, 6 equal-sized medium pots, and 3 equal-sized small pots. The printed instructions for using the fertilizer say: dissolve one capful of fertilizer from the bottle in a liter of water.

Reflecting on the First Problematization

View of a problem posed via the Spontaneous Strategy – A problematization developed through SS (the teacher just helped to direct a situation that started with a dialogue and motivation by the students) led the students to clearly state the problem: “How many groups of four glasses are there in twenty-one and half glasses?” The last interrogative sentence contains the question “How many?” So, we could say that a mathematical problem has been posed because this question can be answered with a number.

View about mathematical content – In my teaching experience, I have seen that students at this level cannot easily interpret a mathematical relation like: “How many x are in y?” Most of them solved the problem using multiplicative reasoning, as follows: “2 X 4, 3 X 4, 5 X 4; there are 5, and a remainder of 1 1/2.” I knew that such an interpretation would require elaborated mathematical reasoning in order to represent it by a division operation, and I observed this behavior in the students.

View of students’ attitudes – The problematization these students engaged in was a very active process and was easily followed by the teacher. The students were very enthusiastic. Only one group showed a lack of interest on the part of most of its members.

A problematization resulting from the use of the “generative theme strategy”.
Generative theme: “the unfinished building”
Group of four students – Pedro, Mário, Adriana and Napoleão – and the teacher

The context for the unfinished building problematization was that the teacher told the students of a previous activity in which some other students investigated the theme of “civil engineering”. Motivated by the dialogue, these students decided to choose as the focus of their theme a nearby building on which construction had stopped for 10 years and which had recently been started again by a new group of workers.

Reflecting on the Second Problematization

View of a problem posed via GTS – Under the GTS and within the theme “the unfinished building”, the students stated the problem: “How much glass would be necessary for the whole building?” As I mentioned before, since the last interrogative sentence contains a question “How much?” we could say that the problematization resulted in a mathematical problem.

View about mathematical content – Despite the fact that the students did not know how to compute using area formulas, they had no difficulty solving this type of area problem. The teacher showed them the meaning of the area of a rectangle and they learned it easily, especially in the group containing the glazier’s nephew.

View about students’ attitudes – All the problematizations using a generative theme were very dynamic and productive. All the students were active and their questions were more mathematically interesting than any of the others.

Conclusions

It is quite evident that students’ interest increases when problematization and problem posing, coming from the facts of social reality, become the heart of instruction in a mathematics class. Given the perceived effects of problematization on the attitudes of the students with regard to mathematical learning that were shown in my research, I have been able to identify four particular advantages, among others, to employing such an approach. They are as follows:

  • Some problematizations will lead students to pose a mathematical problem that introduces a mathematical topic, providing motivation to learn more about the real situation and about the mathematical topic being considered as the students attempt to understand and solve the problem.
  • Some problematization processes will be used essentially to present exercises, material through which to practice skills and techniques.
  • Some problematizations will be used to synthesize what the students have learned, providing them with the opportunity to develop skill in communicating mathematical ideas.
  • Some problematizations and the “possible” problems posed from such processes can develop in students a greater sense of what mathematics is, how it has been created and why they should study it.

Finally, many mathematical concepts were studied by this group of sixth graders, through the interactive method of problematization (during this period referred to above), including, among others: multiplication, division, the composition of multiplicative operators, the distributive property and the mean.

The disadvantages of the problematization process originating in social reality may be explained by the pressures coming from various quarters:

  • Pressure may arise from teachers having an internalist view of mathematics (internalist vs. externalist, as already roughly explained), who just work with questions from inside mathematics.
  • Pressure may arise from the difficulties teachers face in discussing their points of view with others; in other words, from a lack of experience in exchanging ideas.
  • The manner in which time is traditionally structured in school settings does not always allow for the extensive and time-consuming process of problematization so that it can proceed properly.
  • Finally, one needs to consider the potential resistance to any innovative approach that challenges traditional notions of mathematical learning that can come from both parents and students.

References

D´Ambrósio, U. (1986), Da realidade à ação. São Paulo: Editora Summus.

D´Ambrósio, U. (1990). Etnomatemática. São Paulo: Editora Ática.

Dewey, J. (1916). How we think. New York: Macmillan..

Freire, P. (1972). Pedagogy of the oppressed. London: Sheed and Ward.

Freire, P. & Faundez, A. (1986). Por uma pedagogia da pergunta. Rio de Janeiro: Ed. Paz e Terra.

Freire, P. & Shor, I. (1987). Medo e Ousadia. Rio de Janeiro: Ed. Paz e Terra.

Lerman, S. (1989). Investigation: Where to Now? In Paul Ernest (Ed.), Mathematics teaching: The state of the art (ch. 6). Basingstoke: Falmer Press.

Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht: Kluwer.