Teachers researching their practice The experience of the Portuguese group

PDTR Project Teachers researching their practiceThe experience of the Portuguese group Teaching Research in Action July 2006 João Pedro da Ponte, University of LisbonNuno Candeias, Vasco Santana School, Ramada, OdivelasCláudia Nunes, Olivais School, Lisbon

Basics 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: • Posing key questions, • Devising ways of answering those questions in a disciplined way, looking at relevant theories, gathering data, analyzing and interpreting data, • 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 GeometricSketchpad, carrying out investigations and problems? Cláudia Nunes (2004) What do 7th grade pupils think aboutmathematics assessmentand how do they react to innovative practices? Ana Matos (in progress) Does working in problems and exploratory tasks, involvingfunctionscontribute 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, includingsimplifyingalgebraic expressionsandsolving equations. Maria José Molarinho (in progress) Can we develop pupils’ (aged 10) understanding ofrational 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 candeiasan@oniduo.pt Vasco Santana School, Odivelas

Objective How does dynamic geometry software along with explorations, investigations and problems, promote the development of pupils’ geometric competence?

The concept of competenceThe “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 properties Ability to make geometric constructions, namely polygons and locus, that allows the recognition and analysis of their properties. Patterns and investigations Tendency to look for invariants, to explore geometric patterns and to investigate geometric properties and relations. Geometric problem solving Ability to solve geometric problems using constructions and justifying the processes used. Argumentation Ability to formulate valid arguments to justify geometric properties and relations.

Methodology • 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 Topics 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. 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 octagon.

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 difficulties. Other activities

Conclusions 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 cjohnent@yahoo.com.br Olivais School, Lisbon

The problem Objective What do 12-13 year-old pupilsthink of assessment and how they respond to innovative assessment practices? Questions • 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 Assessment 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 skills… • ... 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.

Pedagogicalproposal Principles: Consistent / Diversified / Clear

Research methodology Interviews and questionnaires Qualitative and interpretative methods of analysis Case studies of 4 pupils Investigation about my professional practice 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.

Conclusions 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 processes. 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 • Research questions • Research design - Teaching experiments - Case studies • Research team – colaboration • Data collection – reflective conversation, interview with pupils, researcher journal • Data analysis – role of theory • Dissemination / discussion of research • Concept • 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 Curriculum goals Tasks Exercises Explorations Problems Investigations Strategy Direct teaching Exploratory teaching Evaluation Assessment instruments Assessment modes Evaluation culture Resources Materials ICT-Computers - Calculators

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

3. Research team Colaboration • 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. Supervisor Critical friend Partner Research group Cláudia + Sofia Ana Matos + Neusa Branco …

4. Data collection What is good data? • Provides information about (pupils’, teachers’…) thinking processes, • 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 / Appropriation Purposes • Refine analysis • … Questions • … Findings • Connects to other works Audiences • Research partners • Other teachers • Mathematics educators • Parents • Journalists, General public Web presentation work in progress Cláudia Nunes, Nuno Candeias, Ana Matos, Idália Pesquita, Neusa Branco Papers in professional meetings Cláudia Nunes Articles in professional journals Ana Matos, Neusa Branco, João Pedro da Ponte Papers in research meetings Cláudia Nunes, Nuno Candeias, Idália Pesquita, Neusa Branco Articles in research journals …

7. The concept of researching practice – different meanings Teacher- researcher Action-research Academic research Research about practice Reflection

8. Researching practice– quality criteria Connection with practice it concerns a problem experienced by the actors. Autenticity it expresses the point of view of the actors and its relationship with the social, economic, political and cultural context. Newness 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”).

Teachers researching their practice The experience of the Portuguese group