This paper presents some work developed through a collaborative project involving mathematics teachers and teacher educators, aimed at the development of classroom tasks involving students in mathematical investigations and the study of related teaching styles. We give a general overview of the project and present one task dealing with the concepts of powers and exponents. Then, we describe how it was used in the classroom (grade levels 5-7), and present our experience about the role and the activity of the teacher in the process of organizing, conducting and reflecting about this kind of activity.

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1

Mathematical investigations in the classroom: A collaborative project1

Hélia Oliveira, Universidade de Lisboa

Irene Segurado, Escola B 2, 3 Dr Rui Grácio, Montelavar

João Pedro da Ponte, Universidade de Lisboa

Maria Helena Cunha, Escola Superior de Educação de Viseu

Abstract. This paper presents some work developed through a collaborative project involving

mathematics teachers and teacher educators, aimed at the development of classroom tasks involv-

ing students in mathematical investigations and the study of related teaching styles. We give a

general overview of the project and present one task dealing with the concepts of powers and ex-

ponents. Then, we describe how it was used in the classroom (grade levels 5-7), and present our

experience about the role and the activity of the teacher in the process of organizing, conducting

and reflecting about this kind of activity.

Why mathematical investigations in the classroom?

Modern society requires from everyone a reasonable fluency in mathematics. It

is specially important to be able to interpret information framed in mathematical lan-

guage (numerical and graphical) and to think mathematically (seeking patterns and rela-

tionships and reasoning). However, mathematics is usually regarded as a most difficult

subject. Students frequently view it just as doing computations and getting the correct

answers. They tend to assume a dualist view in which things are either right or wrong.

In many countries, including Portugal, the evaluations of students' learning, attitudes

and views about mathematics are considered unsatisfactory — by all sorts of criteria

(Neves e Serrazina, 1992; Ponte, 1994; Ramalho, 1994).

At the base of these difficulties it is the social filter role played by mathematics

teaching and the subsequent emphasis on mastery of basic concepts and procedures. In

general, mathematics teaching pays little attention to the more advanced aspects of

mathematical activity such as the formulation and resolution of problems, the formula-

tion and testing of conjectures, the pursuit of investigations and mathematical proofs,

and the argumentation and critique of results. While these are fundamental and current

1 Oliveira, H., Segurado, I., Ponte, J. P., & Cunha, H. (1997). Mathematical investigations in the class-

room: A collaborative project. In V. Zack, J. Mousley, & C. Breen (Orgs.), Developing practice: Teach-

ers' inquiry and educational change (pp. 135-142). Geelong, Australia: Centre for Studies in Mathemat-

ics, Science and Environmental Education. (ISBN 0 73002707 4)

2

themes of mathematics education expressed in many curriculum documents across the

world (APM, 1988; Cockcroft, 1982; NCR, 1989; NCTM, 1989) they still find very

little emphasis in classroom practice, both in our country and elsewhere (Lerman, 1989;

Silver, 1993).

Mathematical investigations, based on open-ended problem solving tasks, are

important from the educational point of view (Ernest, 1991; Mason, 1991). In fact, in

our perspective, they:

• are indispensable to provide a complete view of mathematics, since they

are an essential part of mathematical activity;

• stimulate the sort of student involvement required for significant learn-

ing;

• provide multiple entry points for students at different ability levels;

• stimulate a holistic mode of thought, relating many topics, an essential

condition for significant mathematical reasoning.

Setting up a collaborative project

The common feeling that more attention should be paid to understanding what is

involved in doing mathematical investigations in the classroom brought together a small

group of teachers and teachers educators2. We decided to set up a project to experiment

with tasks involving students exploring and investigating around mathematical ideas,

concepts, and processes. Our aim was to engage students in the formulation of conjec-

tures and in mathematical discussion and argumentation, aspects that we find essential

in their mathematical experience. Our work includes producing, experimenting, and

evaluating such tasks and studying the competencies needed for using them to be suc-

cessful in the mathematics classroom.

This activity intends to contribute to furthering knowledge about innovative

learning situations in mathematics classrooms. It is double sided, involving curriculum

development (collecting information about the potential value of given types of activi-

ties in different school grades and acquiring experience in the preparation and evalua-

tion of classroom tasks) and also research on teaching (studying the decisions, dilem-

mas, difficulties, etc. that teachers face in conducting these kinds of activities). Of

2 One of us (João Pedro) has been a classroom teacher for 6 years and afterwards a teacher educator for

16 years. The other three are involved in a master's program. Two of them are classroom teachers who

are now also working in teacher education. One (Hélia) has 4 years of experience in the classroom and

worked for 3 years in teacher education and another (Helena) taught for 5 years in the classroom and has

been in a school of education for 2 years. The other one (Irene) has been a classroom teacher for 16 years.

3

course, our ultimate purpose is to contribute to the dissemination of such activities in the

Portuguese educational system.

The tasks that we are developing are drawn from the mathematics topics of the

2nd and 3rd cycles of basic education in Portugal3, and are designed to fit within the

existing curriculum, and be used in the regular classroom. They are not intended to sug-

gest extra-curricular themes, but to present the topics currently in the curriculum, in a

different way. The tasks do not just stand by themselves. They are organized in units,

each focusing on a single topic. The full set of materials will include worksheets with a

sequence of questions for students, teacher's guides with didactic suggestions and a

supporting text for teachers and students, with relevant information and historical notes

about the given topic. In this paper we present some work based on a worksheet dealing

with the concepts of powers and exponents.

The development of tasks

This project hinges on the collaboration between classroom teachers and re-

searchers/teacher educators4. We start discussing ideas for the tasks in project meetings.

Then, someone from the group produces a first version of a possible worksheet with

three or four questions which may be appropriate for students to follow for two or three

successive lessons. This version is discussed by the whole group in a further meeting

and is usually taken back by the same person to refine. One or two more iterations of

this process usually make the task appropriate for experimentation in the classroom.

Figure 1 presents a set of questions on the topic of powers of natural numbers.

One of our big questions is how much to structure the tasks. Too much structure

implies that we are guiding the students and leaving not enough thinking for them to do.

Too little may imply that the students move in different paths and perhaps miss the ideas

most related to the mathematical concepts in the curriculum. Furthermore, different stu-

dents may profit from being asked different kinds of questions. We finally decided to

make something in between: some of the questions are more structured, others more

open. Often we will begin one worksheet with one or two more structured questions

(such as 1a, 2a, 2b, and 2c) and then provide one or two suggestions for open-ended and

truly exploratory and investigative work (such as 1b and 3).

3 In the Portuguese educational system, this includes grades 5 to 9 (children 10 to 14 years old). In all

these grades mathematics is taught as a separate subject by a specialist teacher.

4 In fact, all four authors are or have been in varying degrees in the roles of classroom teacher and re-

searcher/teacher educator.

4

Figure 1a - Questions on powers and patterns

POWERS AND PATTERNS

Questions

1.-The number 729 may be written as a power of 3. This can be seen making a table with

the powers of 3:

32 = 9

33 = 27

34 = 81

35 = 243

36 = 729

a) Can you write as a power of 2?...

64 =

128 =

200 =

256 =

1000 =

b) What conjectures can you make regarding numbers that can be written as powers of 2?

and as powers of 3?

2.- Look at the following table with powers of 5

51 = 5

52 = 25

53 = 125

54 = 625

a) The last digit of each power is always 5. Is that also true for the next powers of 5?

b) Investigate what happens for the powers of 6

c) Investigate also the powers of 9 and 7.

5

3.- The cubes of the first natural numbers satisfy the following relations:

13 =1

23 =3+5

33 = 7+9+11

• Note that in this example, 13 was written as a "sum" with just one odd number, 23 as a

sum of two odd numbers and 33 as the sum of three odd numbers. Is it the case that the

cube of any number can be written as the sum of odd numbers?

Experimenting and reflecting

Classroom experimentation is carried out by one or more of the teachers in the

group. If possible, one member of the group observes the classes. Sometimes a video

recording is also made. And in some cases we also invite other teachers not in the pro-

ject to try out our tasks. For each experiment, we make a first reflection as soon as pos-

sible after the class and a more extended reflection, using the video recorder if available,

sometime afterwards.

Experiment I. We had in mind that these tasks about powers and exponents could

be used with students of a variety of grade levels. One of the teachers in our group start-

ed experimenting these tasks in a 6th grade class, in a 2-hour period. She regarded this

as a difficult class, since the students had no motivation to study mathematics.

The teacher began by making a short introduction, explained the meaning of the

word "conjecture" and provided the students with an orientation about what they were

going to do. She gave the first page in the first hour (question 1) and students started

working individually. However, they had the opportunity to interact informally with

their peers as they wished. Then, in the second hour, the teacher gave them the second

page (questions 2 and 3). She moved around in the groups, seeing the work of the stu-

dents and discussing their results with them.

Surprisingly for the teacher, the students were very interested and even quite

excited with this activity. This high involvement of the students is illustrated by the fact

that some of them, having finished question 1, started by themselves trying some other

possibilities, such as powers of 5, etc. even without knowing what was to come on the

second sheet (question 2). The teacher felt the experience was a success but commented

that she would like to know more about how to (a) observe the class and (b) evaluate if

all students had understood the same points about the task.

6

Then we had a general reflection in our group about this activity. Focusing on

the issues raised by the teacher, we concluded that observation and evaluation could be

carried out during the students' activity and in a final discussion. Such discussion, in

fact, may be a powerful means of providing all students with further insights about the

work of their peers and giving the teacher a general overview of students' work. But

how to start it? Previously, the teacher felt that after having discussed each task with all

students, one at a time, it did not make much sense to make a final discussion. Also, she

felt that there would be no time to have all the groups reporting their work. The first

group to speak could refer most of the interesting things, leaving little to say to the other

groups. She regarded the discussion as something valuable but difficult to conduct. In

our reflection, a new suggestion come up. It sounds obvious, but the teacher had never

considered it before. Instead of letting each group say all they wanted, she could go task

after task, asking all groups for their results and explanations. Each group, in turn,

would have the opportunity of saying just one thing about its work. The groups that

have less chance of presenting their ideas now may be the first next time.

Other points discussed included how much should the teacher say to the students

while they were working. There was an agreement that the teacher should ask more

questions than provide answers or explanations. We also considered the advantages and

disadvantages of encouraging the students to work in a more collaborative fashion.

Experiment II. A couple of days later this teacher presented the same task to a

5th grade class. When we were designing the questions several times she expressed

doubts about how her 5th graders could handle these but she decided to try anyway. In

this class she asked the students to register everything that they were doing, so that later

she would be able to know what they were thinking, and what conjectures they had

made. The students were working in pairs. In each pair, both students kept separate

notes, but discussion between them was encouraged. The completion of this task took 3

hours (2+1).

The teacher tried a new different strategy regarding students' questions. She was

saying less but asking more questions. This time there was a discussion at the end of the

activity. For each task, one group gave their results and the others reported if they had

something else to add.

Teacher: Let us see now what you have discovered regarding the powers of 5. It

is the turn of AmŽlia's group.

Sílvia: It ends always in 5.

Daniel (a student from another group): And in 2.

Teacher: How is that? Let us hear Daniel. S'lvia says it always ends in 5, right?

But Daniel's group added something else.

7

Daniel: After 25, it ends always in 25.

Teacher: Anything else about the powers of 5?

All the students were very attentive to their colleagues, verifying on the calcula-

tor the results that were given. Their enthusiasm was quite beyond the teacher's highest

expectations.

In the ensuing reflection, we considered the power of these kind of activities to

bring to the teacher new perspectives about her students' capacities. Of course, the

younger kids needed more time to finish the work, but they were able to carry it

through. We also concluded that the strategy used in this experience to make the stu-

dents share results and ideas among themselves represented a big improvement over

what had occurred in the first experiment.

Experiment III. A teacher from another school, not in the project, agreed to try

this task with her 7th graders. This is a teacher who usually presents problem solving

and exploratory activities in her class and often organizes the students in group work.

She began by distributing the full worksheet to the students asking them to set up

groups of 4 or 5. Two hours were devoted to this work plus half an hour of discussion

some days later. Being used to this kind of activity, the students could work productive-

ly by themselves, seldom calling on the teacher.

One interesting episode in this experience is that this teacher at first did not per-

ceive question 3 as interesting. She commented that the question probably had an error

somewhere. But the class was about to begin and there was no time to discuss the issue

in detail. The project member present assured her that the students would be able to

handle it. It was just when they were working on this question that she realized that the

question makes sense, indeed. The students easily realized how they could write 43, 53,

63 ,... as the sum of odd numbers following the pattern presented in the question. Then,

the teacher challenged them to search for a general law. Interestingly, some students had

ideas that we had not thought about when we were designing the task...

The students had no major difficulties working with the questions. Their in-

volvement was high and there were heated discussions within the groups. The students

argued with each other and when they could not agree, they called on the teacher to be

the referee. She avoided giving the answers, challenging the students to find them by

themselves, using expressions such as "experiment", "did you verify?", "try it with an-

other number", "read carefully what is asked"...

The discussion had to be a couple of days later. The teacher presented to the

whole class the thoughts of two of the groups that she found particularly interesting. The

initial involvement of the students was low, since they had forgotten what they had

done. But some of the students became quite intrigued with the approach of one of the

8

groups and asked if that was a mathematical proof. The teacher said that a real proof

would be very difficult to understand, but they were so intrigued that she had a hard

time convincing them of that. She highly enjoyed this since she thinks that the students

should start exploring, justifying, convincing, and proving at an early level.

This teacher considered this experience rather positive and quite consistent with

the activities that she often does in her classes. The students quickly got involved in the

tasks and the fact that they were used to dealing with problematic situations and to make

decisions on their own was probably an important factor. However, looking in retrospect

at this set of lessons, we and the teacher felt that the time elapsed from the activity and

the fact that she presented an initial summary of the student's work made the discussion

much less successful than what it could have been otherwise.

General reflection. At the end of these three experiments, we felt quite strongly

that:

• These tasks are suitable for all students and not just for the better stu-

dents;

• Students can really get involved in significant mathematical activity and

have a sense of doing something significant;

• It is possible that students develop a significant autonomy;

• There is great advantage in stimulating interaction between students,

working in groups or in pairs in these tasks, since a lot of power emerges

from their talking and arguing with each other.

We also concluded that four steps are essential in this kind of activity. In plan-

ning and selecting the tasks the teacher needs to be aware of the mathematical content,

the curriculum goals and capacities, and the previous experience of the students. All of

this must be taken into account and blended into a general perspective of how the work

will proceed. In presenting the activity to the students, the teacher needs to make sure

that they make sense out of it and get really involved. However, not too much must be

said at this point, since the goal is to have the students working on their own. During the

students' activity, the teacher needs to be attentive to successes and problems and be

ready to help if necessary. But we also found that in this kind of activity one must leave

the students to struggle for some time. The role of the teacher is to pose questions and

not immediately tell them all the answers. And we realized that the final discussions are

quite important. The system used with 5th graders worked well. For the students who

have the chance to report first, the discussion may bring to the fore issues and strategies

of which they were not aware. We also found that the discussion should be included in

the activity, or right after if possible.

9

Planning classes together, and observing and discussing them made us aware of

new aspects and issues that we had not consider before. Having some questions in mind,

a process of recording, and an observer in the classroom, were important conditions to

help us in this reflective process. As a group, we are learning about ways of framing

tasks to initiate students' work, to interact with students in doing investigations and in

reflecting about classroom activity. We began choosing the concepts of powers and ex-

ponents as a challenge. At first, this did not seem a very promising topic for our purpos-

es. But, in the end, we are quite pleased with the students' work and with what we have

been learning.

Future work

This work is to be continued in different directions. In terms of curriculum de-

velopment we need to look at issues such as: what is the potential value of given types

of activities in different school years? should the tasks have more or less structure?

should we provide more or less suggestions for teachers? what kinds of suggestions? are

investigations worthwhile in themselves or should they support the learning of mathe-

matical "content"?

Another aspect of concern includes students' learning, where we need to look at

points such as what is the influence of these activities in students' conceptions regarding

school mathematics? What are the relationships between basic mathematical knowledge

and skills and more advanced reasoning processes as conjecturing and proving? How

should we evaluate mathematical processes, and therefore, this kind of activity?

We also need to consider the classroom dynamics: in doing this kind of activity

what issues arise in doing investigations in groups, pairs, individually? what new pat-

terns in classroom interactions are associated with doing mathematical investigations?

And finally, we will be looking further at other teachers, specially to the deci-

sion processes required in conducting these kind of activities: How to integrate these

tasks in the curriculum? How to present them to students? What sort of support to offer

students? How to promote classroom discussions? How to assess students?

For sure, it will not be possible to study all these points at the same level of

depth. But, to conduct classroom investigations, we need to have a clear perspective of

them. Reflecting in the classes where students are encouraged to think and behave

mathematically is important. Considering the growing international literature on this

topic, and discussing these issues with other teachers and researchers outside the group,

we expect to develop and deepen our understandings regarding this kind of activity in

mathematics teaching and learning and challenge others to try it as well in their class-

room practice.

10

References

Associação de Professores de Matemática (1988). Renovação do currículo de Matemá-

tica. Lisboa: APM.

Cockcroft, W. H. (1982). Mathematics counts. London: HMSO.

Ernest, P. (1991). The philosophy of mathematics education. London: Falmer.

Lerman, S. (1989). Investigations: Where to now? In P. Ernest (Ed.), Mathematics

teaching: The state of the art (pp. 73-80). London: Falmer.

Mason, J. (1991). Mathematical problem solving: Open, closed and exploratory in the

UK. ZDM, 91(1), 14-19.

National Council of Teachers of Mathematics (1989). Curriculum and evaluation

standards for school mathematics. Reston: NCTM.

National Research Council (1989). Everybody counts: A report to the nation o the fu-

ture of Mathematics Education. Washington: National Academy Press.

Neves, L., & Serrazina, L. (1992). O desempenho em matemática aos 9 e 13 anos. Edu-

cação e Matemática, 22 , 26-28.

Ponte, J. P. (1994). Uma disciplina condenada ao insucesso? NOESIS, 32, 24-26.

Ramalho, G. (1994). As nossas crianças e a Matemática: Caracterização da participa-

ção dos alunos portugueses no Second International Assessment of Educational

Progress. Lisboa: DEPGEF.

Silver, E. A. (1993). On mathematical problem posing. Proceedings of the XV PME

Meeting, Vol. I, pp. 66-85. Tsukuba, Japan.

... Indeed, investigations are experienced with authenticity when students are confident enough to pose their own problems and take their own initiative. Investigations help students to develop significant autonomy, something that stimulates interaction among students (Oliveira et al., 1997). Furthermore, as students engage actively with such tasks, misconceptions can be revealed and this can assist the classroom assessment process. ...

... identify suitable tasks that match the ability and interests of the students (Greenes, 1996;Ponte et al., 1998;Ponte, Segurado & Oliveira, 2003). It follows that teachers need to be aware of the previous learning experiences of the students (Oliveira et al., 1997). For students can find meaning and are able to take up the challenge when they can start working on the task from their present state of knowledge. ...

... It becomes crucial at this stage to discuss the roles of the students and their teacher during an investigation. Studies (e.g., Orton & Wain, 1994;Oliveira et al., 1997;Van Reeuwijk & Wijers, 2004;Doerr, 2006) have shown that students encounter difficulties adjusting to when the set tasks lack structure. To help them adjust, students can be presented with -term goals as these offer immediate satisfaction and feedback. ...

... A respeito da Matemática, percebe-se que essa ainda se concentra, basicamente, no teórico e abstrato, privilegiando manipulações algébricas na busca pela resposta correta. Nesse cenário, é dado pouca (ou nenhuma) atenção a aspectos relevantes dessa ciência, como a formulação e a resolução de problemas, a elaboração e o teste de conjecturas, a participação em investigações, a busca por provas matemáticas, a argumentação e a análise crítica de possíveis resultados encontrados (Oliveira, Segurado, Ponte & Cunha, 1997). quais, não possua uma resposta pronta, de forma que passe a procurá-la, tanto quanto possível, de modo matematicamente fundamentado (Ponte et al., 2016). ...

... A investigação matemática é tida como indispensável por Oliveira et al. (1997) para promover uma visão completa da disciplina, uma vez que se caracteriza como uma parte essencial da atividade matemática. Levando-se em consideração que "não é óbvio como o professor pode articular a realização de investigações com outros tipos de atividades" (Ponte et al., 2016, p ...

Este artigo relata uma pesquisa sobre a utilização da prática da investigação matemática em aula, por meio da Modelagem Matemática, no contexto de ensino e aprendizagem de equações diferenciais ordinárias de variáveis separáveis. Os sujeitos envolvidos são alunos da Licenciatura em Matemática de uma instituição pública de ensino em Alegrete, RS. Na fundamentação teórica destaca-se a Modelação como método de ensino e aprendizagem com pesquisa, que busca a contextualização e o protagonismo do estudante. Para tanto, aplicou-se um conjunto de atividades que visou a construção do clássico modelo designado de Lei de resfriamento de Newton, além de um breve questionário, cujas respostas foram estudadas mediante a Análise Textual Discursiva. Os discentes foram instigados pelo método adotado e mostraram satisfação com a possibilidade da união da teoria com a prática, enaltecendo a contextualização dos conceitos estudados.

... Mathematical investigations are also important from the educational point of view. In our perspective (Oliveira et al., 1997), they: (a) stimulate the sort of student involvement required for significant learning; (b) provide multiple entry points for students at different ability levels; (c) stimulate a holistic mode of thought, relating many topics, a basic condition for significant mathematical reasoning; and (d) are indispensable to provide a complete view of mathematics, since they are an essential part of mathematical activity. ...

There are good reasons why we may involve the students in doing mathematics investigations. Recent curricula encourage this sort of activity but we notice that its application in the classroom is not a simple matter. We discuss the issues that arise when students are presented with investigative tasks, with special interest in the dynamics of the classroom and in the role of the teacher. Our aim is to derive suggestions for classroom practice as well as for further research and teacher development.

... At the same time, a considerable number of teachers are following post-graduate studies on education, in particular on various aspects of the didactics of mathematics, and some of them are participating in projects which involve both a strong component of curricular innovation and a research dimension. Since the creation of APM, and following the experience of the MAT789 project (Abrantes, 1993), the cooperation between teachers and researchers became an interesting feature of Portuguese mathematics education -see, for example, Oliveira et al. (1997), Ponte et al. (1998) or Porfírio and Abrantes (1999). It is not surprising that curricular innovation and teachers professional development became two major areas of research. ...

  • Paulo Abrantes

A movement of educational change has been developed in Portugal, aiming at giving the schools a larger autonomy in curricular decisions. In reconceiving the view about the curriculum, the concept of "competence" plays a central role while the process of innovation constitutes a major aspect. This movement is described and analysed, in particular by discussing the notion of competence and the characteristics of the process of curriculum development. A special focus is on the way in which mathematical competence for all may be interpreted and how it is related to developments in mathematics education. The analysis of obstacles emerging in a large-scale educational change may be relevant for discussion in an international context and offers some suggestions for future research and debate. 1. New trends in the Portuguese educational system

... However, there is an urgent need for classroom-based research that addresses the teaching and learning issues associated with young children undertaking investigations. While classroom-based research on the teaching and learning of children conducting mathematical investigations exists for the upper primary years (e.g., Oliveira, Segurado, da Ponte, & Cunha, 1997), there is limited research in the early primary years. The existing research on young children implementing investigations tends to focus on the implementation of an investigation with an individual child (e.g., Juraschek & Evans, 1997), and hence, provides scant guidance for implementing investigations with a class. ...

One approach to providing a mathematically rich curriculum is to involve young children in mathematical investigations in which they engage in the exploration of meaningful problems, and problem posing. However, there is limited research on how teachers can facilitate young children's learning through investigations. This study explored the difficulties seven-to-eight year old students experienced when they began an investigatory program. We present examples of specific difficulties students confronted in conceptualising and conducting investigations, as well as general difficulties that they experienced which hindered their investigations, such as limited observation skills. Our contention is that mathematical investigations can enhance young children's learning provided that their difficulties are addressed.

... Additionally, through investigations, children gain insight into cultural practices of mathematicians, and mathematics as a career (National Council of Teachers of Mathematics [NCTM], 2000). Investigations represent radically new practices (Klinman, Russell, Wright, & Mokros, 1998;Taber, 1998) and while research exists on ways teachers can support older students' investigations (e.g., Greenes, 1996;Oliveira, Segurado, da Ponte, & Cunha, 1997;Taber, 1998), research with young children appears to be limited to descriptions of individual children's learning (e.g., Whitin, 1993) and classroom mathematics programs (e.g., Skinner, 1999;Whitin, 1989). Given the importance attributed to investigations, research is urgently needed to explore the learning issues that will confront teachers who are implementing investigations in their classrooms. ...

Engaging children in mathematical investigations is advocated as a means of facilitating mathematical learning. However there is limited guidance for teachers on ways to support young children engaged in investigations. This study provides insights into the mathematical literacy required by seven-to-eight-year-old students undertaking investigations. Examples of difficulties are described in relation to problem solving, representation, manipulation, and reasoning. While mathematical investigations can enhance young children's learning, teachers need to provide guidance to address necessary skills and knowledge.

O texto apresentado é um recorte de uma pesquisa de mestrado e explicita a articulação revelada entre a Investigação Matemática e a História da Matemática, as Tecnologias da Informação e Comunicação - TIC's e a Resolução de Problemas, em registros textuais de professores. A metodologia de pesquisa é qualitativa pautada nos pressupostos do método fenomenológico-hermenêutico. A articulação mostrou-se teoricamente possível, todavia qualquer tentativa de inter-relacioná-las requer clareza das particularidades, das regiões de inquérito, da epistemologia e dos objetivos de ensino de cada uma das Tendências, caso contrário esse caráter interlocutor pode configurar uma anomalia que afeta o núcleo firme da Investigação Matemática no tocante ao princípio da natureza das suas tarefas, além de gerar conflitos epistemológicos e de aprendizagem.

O texto apresentado é um recorte de uma pesquisa de mestrado e explicita a articulação revelada entre a Investigação Matemática e a História da Matemática, as Tecnologias da Informação e Comunicação - TIC's e a Resolução de Problemas, em registros textuais de professores. A metodologia de pesquisa é qualitativa pautada nos pressupostos do método fenomenológico-hermenêutico. A articulação mostrou-se teoricamente possível, todavia qualquer tentativa de inter-relacioná-las requer clareza das particularidades, das regiões de inquérito, da epistemologia e dos objetivos de ensino de cada uma das Tendências, caso contrário esse caráter interlocutor pode configurar uma anomalia que afeta o núcleo firme da Investigação Matemática no tocante ao princípio da natureza das suas tarefas, além de gerar conflitos epistemológicos e de aprendizagem.

  • Irene Segurado
  • João Pedro da Ponte João Pedro da Ponte

As actividades de investigação proporcionam aos alunos uma experiência viva e gratificante – levando-os a aprender processos como generalizar, considerar casos particulares, simbolizar, comunicar, analisar, explorar, conjecturar e provar. Investigação anterior, embora escassa, mostra as suas potencialidades mas t6ambém os seus problemas, resultantes dos alunos manifestarem, frequentemente, concepções incorrectas sobre a Matemática e a sua aprendizagem. Este estudo procura saber como é que eles trabalham nestas actividades e de que forma podem evoluir as suas concepções. Para isso – tendo por base a realização de quatro tarefas de investigação em aulas de Matemática e usando um estudo de caso – analisamos o modo como um alunos do 6º ano de escolaridade se envolve em actividades deste tipo. O aluno em causa, Francisco, mostra grande interesse nas actividades propostas. Nas primeiras não vai além da formulação de conjecturas. Progressivamente, realiza testes, refina conjecturas e ensaia justificações. Revela crescente autonomia, confiança e ousadia nos seus raciocínios. De início, integra-se pouco no grupo, mas por fim já interage bastante com os colegas. Inicialmente muito dependente da professora para a validação dos resultados, vai reconhecendo que também é uma autoridade matemática. Estas actividades contribuem para que ele adquira uma nova visão da Matemática como ciência em desenvolvimento, do papel do professor como orientador e do carácter desejavelmente estimulante das tarefas. O estudo conclui que a natureza desafiante e aberta das tarefas, o modo de trabalho usado pela professora e a dinâmica da aula proporcionaram oportunidades de raciocínio e interacção que foram aproveitadas por este aluno. Mostra também que é possível um significativo enriquecimento de aspectos cruciais das concepções dos alunos e sugere que o trabalho investigativo na sala de aula merece uma grande atenção na pesquisa educacional, indiciando novas pistas para trabalho futuro. Abstract. Investigative activities provide students with lively and gratifying experiences – stimulating learning processes such as generalizing. Particularizing, symbolizing, communicating, analyzing, exploring, conjecturing, and proving. Previous research, although scarce, points their problems. These result from incorrect conceptions that students often hold about mathematics and learning. This research intends to show how they work in these activities and develop their conceptions. Based in four investigational tasks and using a case study it analyses how a sixth grade student gets involved in this kind of activity. The student, Francisco, shows great interest in the proposed tasks. Firstly, he does not go beyond formulating conjectures. Progressively, he carries out tests, refines his conjectures, and sketches justifications. He reveals growing autonomy, confidence, and willingness to take risks in his reasoning. In the beginning, he does not relate to the group, but at the end he interacts with his colleagues. Initially very dependent from the teacher to validate results, he recognizes himself as a mathematical authority. These activities 1 Segurado, I., & Ponte, J. P. (1998). Concepções sobre a matemática e trabalho investigativo. Quadrante, 7(2), 5-40.

  • Paul Ernest Paul Ernest

It is suggested that the philosophy of mathematics is relevant to mathematics education (1) because the philosophical schools of thought have a direct bearing on educational issues and (2) because new entrants to teaching may bring with them undiluted theoretical views on the nature of mathematics.The views of the following five schools of thought are sketched: Logicism, Formalism, Constructivism, Platonism and Falliblism. Subsequently the relationship between these views and certain current issues in mathematics education is discussed. It is concluded that each of the views provides insights as to the nature of mathematics, but that Falliblism is perhaps the only viewpoint compatible with humane mathematics education.

  • Paul Ernest Paul Ernest

L'A. repond a la critique que porte A. Gowri a son propre ouvrage dans un article intitule «The Philosophy of Mathematics Education by P. Ernest», en precisant sa position a l'egard des mathematiques, de l'elitisme et de l'enseignement a un public plus large, tout en soulignant la contradiction inherente a cette discipline qu'il tente de resoudre par la conception d'une ethnomathematique

  • Paul Ernest Paul Ernest

The central concern of the philosophy of mathematics is to give an account of the nature of mathematics. Views of the nature of mathematics are particularly important in the teaching of mathematics, where they can strongly influence the mathematics curriculum as taught to pupils. However, the distinction must be drawn between stated beliefs as to the nature of mathematics and views as inferred from actual classroom practice. In mathematics education there is increasing emphasis on the process (as opposed to the product) view of mathematics. This orientation is snared by a new wave in the philosophy of mathematics, represented in the works of Imre Lakatos, which rejects the traditional product orientation of the philosophy of mathematics for deep philosophical reasons. Thus there is a growing school of thought in the philosophy of mathematics which is able to account for the nature of mathematics in a way which is fruitful for the philosophers, educationists, teachers and students of mathematics alike.

  • Cooper
  • Judith M

Typescript. Thesis (Ph. D.)--George Mason University, 1996. Includes bibliographical references (leaves 135-148). Vita: leaf 188.

Investigations: Where to now?

  • S Lerman

Lerman, S. (1989). Investigations: Where to now? In P. Ernest (Ed.), Mathematics teaching: The state of the art (pp. 73-80). London: Falmer.

Mathematical problem solving: Open, closed and exploratory in the UK. ZDM

  • J Mason

Mason, J. (1991). Mathematical problem solving: Open, closed and exploratory in the UK. ZDM, 91(1), 14-19.

O desempenho em matemática aos 9 e 13 anos

  • L Neves
  • L Serrazina

Neves, L., & Serrazina, L. (1992). O desempenho em matemática aos 9 e 13 anos. Educação e Matemática, 22, 26-28.