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					                   PDTR Project

Teachers researching their practice

The experience of the Portuguese group

        João Pedro da Ponte, University of Lisbon
  Nuno Candeias, Vasco Santana School, Ramada, Odivelas
          Cláudia Nunes, Olivais School, Lisbon

          Teaching Research in Action
                   July 2006
What is a teacher researcher?
  A teacher researcher is a teacher who does research, usually
   concerning problems of his/her own professional practice.
What is research?
  An activity that involves:
    1.Posing key questions,
    2.Devising ways of answering those questions in a disciplined
      way, looking at relevant theories, gathering data, analyzing and
      interpreting data,
    3.Presenting its results and ideas, sharing concerns with the
      relevant research community.
                                        (Beillerot, 2001; Ponte, 2002)

        A teacher researcher reflects on his/her own practice…
           BUT not every teacher that reflects is a researcher!...
          Why teachers do research?
In teaching, assessing, participating in the school activity, and
working with the community…
   Teachers face many problems in professional practice…
   And want to find better ways of dealing with them.
Research is a process of constructing knowledge about practice, to
   The teachers involved,
   Other teachers (of the same school and of other schools),
   Other communities (including academic) and the society at large.
Teachers carry out research on their practice
   To become main agents in the curriculum and professional fields, with
    more powerful resources to face problems,
   As a key form of professional/organizational development,
   Contributing to the culture and knowledge base of the profession,
   As a contribution to the knowledge of educational problems.
 Examples of research questions
Nuno Candeias (2005)
 How do 8th grade pupils (aged 13) develop their geometrical competence
 when they use Geometric Sketchpad, carrying out investigations and problems?
Cláudia Nunes (2004)
 What do 7th grade pupils think about mathematics assessment and how do
 they react to innovative practices?
Ana Matos (in progress)
 Does working in problems and exploratory tasks, involving functions contribute
 to the development of algebraic thinking, in 8th grade pupils?
Idália Pesquita (in progress)
 What are the reasoning processes and difficulties of 8th grade pupils when they
 work with situations that require algebraic thinking, including simplifying
 algebraic expressions and solving equations.
Maria José Molarinho (in progress)
 Can we develop pupils’ (aged 10) understanding of rational numbers, using a
 exploratory strategy, daily life situations, and the empty line representation?
Teaching 8th degree pupils (13-14 year-
 old) with dynamic geometry software

              Nuno Candeias
       Vasco Santana School, Odivelas

How does dynamic geometry software along with
explorations, investigations and problems, promote the
development of pupils’ geometric competence?
       The concept of competence
         The “Portuguese” meaning

The culture that everybody must develop as a
consequence of his/her basic education that supposes
the acquisition of knowledge and appropriation of a set
of basic procedures but does not identify with the
memorized knowledge of terms, facts, and basic
procedures, without the elements of understanding,
interpretation, and solving problems.
                                    (ME-DEB, 2001, p. 9)
             Geometric competence

Construction of figures and analysis of their
 Ability to make geometric constructions, namely polygons
 and locus, that allows the recognition and analysis of their
Patterns and investigations
 Tendency to look for invariants, to explore geometric
 patterns and to investigate geometric properties and
Geometric problem solving
 Ability to solve geometric problems using constructions and
 justifying the processes used.
 Ability to formulate valid arguments to justify geometric
 properties and relations.
• Qualitative research
• Teacher research
  – a teacher researching his own practice.
• Case study
  – three groups of two pupils each.
• Data sources
  –   interviews;
  –   investigator’s diary;
  –   pupil’s written answers;
  –   initial and final questionnaire.
• Analysis-Categories
  –   construction of figures and analysis of their properties;
  –   patterns and investigations;
  –   geometric problem solving;
  –   pupil’s conceptions about geometry.
              Pedagogical proposal
Angles, triangles, quadrilaterals, symmetry axes;
Decomposition of figures and Pythagoras theorem; Locus;
Translation and Similarity of triangles.
Working plan
9 groups of 2 pupils each; 26 classes of 90 minutes;
1 informatics classroom; 26 activities.
Explorations (11); open problems (7); problem solving (8).
Pupil’s evaluation
Work done in the classroom; activities 8, 9 e 17 (open
problems report) and activities 21 e 26 (problem solving);
homework; attitudes in the classroom.
         Pupils in geometry classes
                (Activity 24 - open problem)

Teacher: Well, have you answered the last question?
José: Yes! We can tile it with squares and rectangles.
Teacher: Why?
José: We made translations with them to cover everything.
Teacher: Have you tried tiling with triangles?
José: We could tile with equilateral triangles, but we needed
 to rotate some of them, therefore it wasn’t translations.
Teacher: All right! And with rhombus and kites?
José: With rhombus we can tile but with kites we can’t. I
 think it’s related with symmetry axes.
Teacher: Why?
         Pupils in geometry classes
                 (Activity 24 - open problem)

José: The square has 4, the rectangle has 2, the rhombus has
 also 2 and the kite has just 1.
Teacher: What about the parallelogram, can you tile with it?
José: Yes and it has 0 symmetry axes.
Teacher: What conjecture can you write?
José: I think that when the number of symmetry axes is even
 we can tile with translations.
Teacher: This conjecture is interesting! We need to
 demonstrate it to see if the conjecture is true, or to find an
 counterexample to say that is false.
         Pupils in geometry classes
                (Activity 24 - open problem)

• Some minutes later José called me again to say that the
  hexagon had 6 symmetry axes and he could tile with it;
• He was convicted that the conjecture was true because
  he didn’t find a counterexample;
• They wrote:
 “It’s impossible to make a tiling with triangles using the menu
 Translate. With quadrilaterals, we can do a tiling with
 rectangles, squares and rhombus because they have an even
 number of symmetry axes. Furthermore, all polygons with an
 even number of symmetry axes can tile the sketch.”
• At night I investigated with Sketchpad and, in the next
  class, I asked them to see what happens with the
       Pupils in geometry classes
               (Activity 21– problem solving)

3rd Problem:

  In a basketball game the ball is 4 m from
  Manuel and 5 m from Sara. Where is the ball?
           Pupils in geometry classes
                   (Activity 21– problem solving)
Initial solution
presented by
the pupils:
          Pupils in geometry classes
                (Activity 21– problem solving)

After my remark (suggesting them to start by drawing
the positions of the people and then finding the different
positions of the ball) the pupils solved the problem again
and wrote the following conclusions:

(i) More than 9 m, there is no solution (the circumferences do
not intersect);
(ii) Less than 9 m and more than 1 m, the ball can be in two
different places (intersection points of the circumferences);
(iii) Exactly at 1m, the ball can be in just one place (point of
tangency the circumferences);
(iv) Less than 1 m, there is no solution (the circumferences
do not intersect).
       The class in geometry lessons
Time spent vs. time planned
Assessment of open problem
• Global assessment consistence;
• More difficulties in communication of results;
• Better in mathematical knowledge;
Assessment of problem solving
• Evaluation of each problem,
• Difficulties understanding the problem,
• Verification of the solution found,
• The problem was not read again,
• Some strategies were incomplete,
• Writing of resolution processes.
      The class in geometry lessons

General discussions
Presenting results to team mates in some activities.
José’s commentaries in the interview
Those discussions are useful for us to see things that we
couldn’ t see and other ways of solving it. Sometimes we can’t
solve it, so those discussions are useful to understand our
Other activities
1. Construction of figures and analysis of their properties.
Successfully developed by most pupils; clear answers despite writing
difficulties; Sketchpad’s role.
2. Patterns and investigations. Pupils’ resolution of an activity
during the interviews; investigative spirit; follow their own path;
Sketchpad’ s role.
3. Geometric problem solving. Challenges; differences to the
open problems activities; learning impact; reformulation of answers
and solving processes; easer to begin from scratch.
4. Pupils’ perform in the different aspects of the geometric
competence. Unequal development; different performances vs.
constant performances.
5. Pupil’s conceptions about geometry. No longer identified
with a specific topic:
                      “challenges to overcome”
   “finding concepts, making relations and obtaining conclusions”
Assessment as a regulatory process in
 mathematics teaching and learning

   A study with 12-13 years old pupils

          Cláudia Canha Nunes
            Olivais School, Lisbon
                 The problem

What do 12-13 year-old pupils think of assessment and
 how they respond to innovative assessment practices?
 How do pupils regard assessment and how do they
  engage in it?
 How do they perceive the formative and regulatory
  goals of assessment?
 How do they respond to different assessment tools
  and methods and how do these shape their
  conceptions of assessment and of mathematics?
             Theoretical framework
Formative and regulatory function of assessment of pupils’ skills...
   ... Consistent with different practices, there were different
    evaluation tools and methods,

Pupils should have an active participation in assessment...
   ... Through the negotiation of assessment criteria, the regulation of
    practices and self-evaluation,

Teacher feedback can help pupils to learn and improve their
   ... Because it gives them useful information about their difficulties
    and progresses.

Abrantes, 2002 / APM, 1998 / Hadji, 1994 / Leal, 1992 / NCTM, 1999
             Pedagogical proposal

Assessment contract/culture
  Teacher’s feedback to pupils and pupils’ feedback to the teacher,
  Assessment with negotiation and pupils’ participation.

Collaboration with another teacher (Sofia)
  To avoid professional isolation,
  To dialogue, to reflect and to learn.

Activities with parents
  Active agents in pupils’ attendance,
  Essential allies for well succeed work.
          Pedagogical proposal
                  Test I                            Test II
      Portfolio    /     Synthesis     /    Self oral evaluation

   January        February     March        April     May          June

Task I Task II     Task III    Task IV     Task V   Task VI

                       Project work

          Principles: Consistent / Diversified / Clear
  Research methodology
                Qualitative and interpretative
                    methods of analysis

Case studies                                Investigation about my
  of 4 pupils                               professional practice

  Interviews and
                                          My diary

   Bodgan e Biklen, 1994 / Ponte, 1994, 2002 / Yin, 1989
         Activities with parents

The first meeting was essential...
  ... To explain the pedagogical proposal and the
   assessment contract,
  ... To involve the parents in the assessment process and
   to be responsible for the pupil’s work and learning.

       I agree with your pedagogical proposal because is
        important to change and make something different.
     Pupils’ results prove that the traditional method has failed.
          Activities with parents

The second meeting was important...
  ... To make a set reflection of this succeful work,
  … To strengthen the logical necessity of making
   continuous this assessment culture in an atmosphere of
   dialogue and mutual support between teachers, pupils
   and parents.

     As parents we can ask that some of these assessment
        instruments are used in the school in the future.
Different assessment tools and methods
  Research reports / Project work / Portfolio / Two fases test /
   Synthesis / Self oral evaluation
Perception of formative and regulatory function of assessment
  The scaffolding and feedback given to pupils throughout the school
   year helped them learn and it was an important step to envolve them
   in the regulation of this process.
Engagement in learning and assessment process
  The instituted culture of assessment reduced their feelings of anxiety
   towards assessment practices.
Improvement in pupils’ conceptions
  An atmosphere of dialogue and mutual support between teachers,
   pupils and parents can contribute to an improvement in pupils’
   conceptions regarding assessment and mathematics.
                    Final reflection
Curriculum management and assessment
  To balance different tasks and methods,
  To select correct tasks,
  To reflect after the mathematics activity,
  To know my pupils,
  To make use of assessment to regulate the teaching and learning
Assessment contract/culture
  From theoretical principles (consistent, diversified, clear)...
  ... To the daily diligence to make it real.
Collaboration with Sofia
  Emotional support and creative professional factor,
  Professional improvement and enrichment.
     Issues in teacher research

1.   Research questions
2.   Research design - Teaching experiments - Case
3.   Research team – colaboration
4.   Data collection – reflective conversation, interview
     with pupils, researcher journal
5.   Data analysis – role of theory
6.   Dissemination / discussion of research
7.   Concept
8.   Quality
   1. Examples of research questions
It is relevant to inform practice
 Focus on important mathematical topics: Rational Numbers (Mª
    José), Algebraic processes (Idália), Functions (Ana), Geometric
    competence (Nuno),
 Deal with important professional issues: Assessment (Cláudia),
  Curriculum strategy (Mª José, Nuno).
… possible to get empirical evidence to respond
 Collecting data in the classroom and from our own pupils.
… linkable to theory
 Algebraic thinking (Ana, Idália); Geometric thinking (Nuno); Number
  learning (Mª José),
 Pupils’ conceptions (Cláudia, Nuno),
 Curriculum, tasks and activity (all); ICT in mathematics education
  (Nuno); Assessment in mathematics education (Cláudia),
 Communication in the classroom (Carmen, Sílvia).
   A look at mathematics teaching

 Mathematics                Tasks
Curriculum goals           Exercises
                                                Direct teaching
     Evaluation                               Exploratory teaching
Assessment instruments
  Assessment modes
   Evaluation culture
                                ICT-Computers - Calculators
                          2. Research plan
    Teaching Experiment
     Unit / Principles /Tasks – Strategy – Pupils’ evaluation.

    Case studies
     Analyze complex objects that may be seen as a unit.

    Investigating our own practice
    • Distance researcher / object.

   Teaching plan prep.              Teaching        Collecting   Analysing
(col. baseline information)   (col. ongoing data)   extra data     data
                    3. Research team

     Joining the efforts of several people in solving a single
      problem, is a very helpful strategy to face problems of
      professional practice.
     Several people working together
        Have more ideas, more energy and more strength to
         overcome obstacles than a single person,
        May draw on individual competences,
        ... However, they need to adjust to each other, learning
         how to work efficiently with each other.

Critical friend                         Cláudia + Sofia
Partner                                 Ana Matos + Neusa Branco
Research group                          …
                4. Data collection

What is good data?
 Provides information about (pupils’, teachers’…) thinking
 Is rich in meanings and negotiation of meanings.

Instruments / forms of data collection
 Interviews with pupils (Ana, Cláudia, Idália, Nuno),
 Researchers’ journal, with classroom reports and
  reflections (Ana, Cláudia, Idália, Nuno),
 Gathering pupils written work (Ana, Cláudia, Idália, Nuno),
 Questionnaires (Cláudia, Nuno).
               5. Data analysis
Qualitative / Interpretative

 Focus on meanings, understandings, explanations,
 Seeking to understand things from the point of view
  of the participant.

Categorization – Interpretation

 Emergent categories, based in research questions
  and in relevant theory,
 Construction of meaningful narratives for
  professional and mathematics education audiences.
       6. Dissemination / Discussion /
Purposes                        Web presentation work in progress
 Refine analysis               Cláudia Nunes, Nuno Candeias, Ana
 … Questions                   Matos, Idália Pesquita, Neusa Branco
 … Findings                    Papers in professional meetings
 Connects to other works       Cláudia Nunes

                                Articles in professional journals
Audiences                       Ana Matos, Neusa Branco, João Pedro
 Research partners             da Ponte
 Other teachers
                                Papers in research meetings
 Mathematics educators         Cláudia Nunes, Nuno Candeias, Idália
 Parents                       Pesquita, Neusa Branco
 Journalists, General public
                                Articles in research journals
7. The concept of researching practice
         – different meanings


 research       Research about

                   Teacher- researcher
          8. Researching practice
              – quality criteria
Connection with practice
 it concerns a problem experienced by the actors.
 it expresses the point of view of the actors and its relationship with
 the social, economic, political and cultural context.
 it has new elements, in formulation questions, in the methodology
 used, or in the interpretation of the results.
Methodological quality
 it has explicit questions and procedures of data collection and
 presents the conclusions based in the evidence collected.
Dialogic quality
  it is public and discussed by other actors (“close” and “distant”).

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