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PDTR Project. Portuguese National Report. João Pedro da Ponte , University of Lisbon Nuno Candeias, Vasco Santana School, Ramada, Odivelas Cláudia Nunes, Olivais School, Lisbon Ana Matos, Gama Barros School, Cacém, Sintra Idália Pesquita, D. Carlos I School, Sintra

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## Portuguese National Report

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**PDTR Project**Portuguese National Report João Pedro da Ponte, University of LisbonNuno Candeias, Vasco Santana School, Ramada, OdivelasCláudia Nunes, Olivais School, Lisbon Ana Matos, Gama Barros School, Cacém, Sintra Idália Pesquita, D. Carlos I School, Sintra Maria José Molarinho, Gaspar CorreiaSchool, Portela, Loures July 2006**Who is who**Teachers Ana Isabel Silvestre, Cristina Garcia, Guida Rocha, Isilda Marques, Maria José Molarinho, Sandra Marques, Sara Costa, Xana Simões (5th-6th grade / 10-11 years old) Ana Matos, Carmen Salvado, Elisa Mosquito, Idália Pesquita, Maria João Lagarto, Neusa Branco, Sílvia Dias (7th-9th grade / 12-14 years old) Mentors João Pedro da Ponte, Hélia Oliveira (University teachers at Department of Education, FCUL) Cláudia Nunes, Nuno Candeias (secondary school mathematics teachers) English teacher Rosário Oliveira (secondary school English Teacher, temporarily at the Ministry of Science, Innovation and Technology). Mathematics coordinator António Domingos (University teacher at Departament of Mathematics, FCT-UNL)**Learning by looking at selected examples**Activity in the three strands Teacher researcher (João Pedro da Ponte, Cláudia Nunes, Hélia Oliveira, Nuno Candeias) • 14 sessions (four hours), including • Introduction and reflection; • Discussion of projects: Ana Isabel Silvestre, Neusa Branco, Ana Matos, Cinta Muñoz, Idália Pesquita, Sandra Magina. • Discussion of papers: João Almiro; • Mini study; English (Rosário Oliveira) • 5 sessions (four hours), devoted to oral and writen work dealing with English Language Mathematics (António Domingos) • 5 sessions (four hours), each one devoted to the exploration of a mathematics field**PISA Mini-study**• Preparation • Organizing target populations (pupils aged 11 and 13-14) • Organizing teams • Selecting instruments Objectives To know the reasoning strategies and dificulties of Portuguese pupils in • Algebra: Patterns, Simbolization and Change • Geometry: Vizualization, Area, Proportional reasoning • Interpretation of results and reflection on implications • Transcribing and organizing data • Writting a detailed report • Selecting issues to present PDTR • Making two presentations at a Math Eduction research meeting • Data collection • Writen test • Interviews • (common items)**Reasoning strategies and difficulties of pupils in algebra**Ana Matos, Carmen Salvado, Cláudia Nunes, Idália Pesquita, Elisa Mosquito, Hélia Oliveira, Maria João Lagarto, Neusa Branco e Sílvia Dias**Theoretical framework - algebra**Change presupposes • Use and interpret different representations: symbolic, algebraic, graphics, lists and geometrics, • Contact with algebra. Patterns and regularities • Recognize patterns and regularities, • Solve problems involving shapes and patterns, • Describe a patterns using algebraic language. Use and interpretation of symbolic language • Understand the meaning of symbols and letters, • Work with the equivalence principles. NCTM, 2000**Methodology**Descriptive and interpretative Qualitative and interpretative Written test with 15 questions (94 pupils) 101 pupils 13-15 years old Interview with 10 questions (7 pupils) Focus on algebra**190**180 170 160 150 140 130 190 Height (cm) Average height of young males - 1998 Average height of young females - 1998 Age (Years) 10 11 12 13 14 15 16 17 18 19 20 Task I – Growing up In 1998 the average height of both young males and young females in the Netherlands is represented in this graph. 1.2. Since 1980 the average height of 20-year-old females has increased by 2.3 cm, to 170.6 cm. What was the average height of a 20-year-old female in 1980? Item type: Overarching idea: Close-constructed response Change and relationships Competency cluster: Situation: Reproduction Scientific**Strategy 1**The pupil reads the question and see that they can solve it using the given information. D- [Diana thinking] The height is 170.6 cm. Ok, I know what to do next! I have to subtract 2.3 cm from 170.6 cm. T- This will give you the… D- Average height! T- Correct. D- [Diana is writing in scrap paper the computations she is making in her head] The result is 158.3 cm. T- Do you get the same result when you use the calculator? D- Aha…Oops ok, I get it. The value is 168.3 cm. In 1980, 20-year-old girls measure 168.3 cm Diana Diana identifies the question, but gets the mental computation wrong. She uses the calculator to verify her result and changes it accordingly.**Strategy 2**Pupils read the question, but they consider that the information is not enough to answer the question, so they choose to read the graph. Joana... J- So let us see, in 1980 the average height for 20 year-old person ... [points to the graph] T – This graph pertains to...? J – To 1998. T – Then how will we solve this problem? J – If 2.3 cm, 1.70 m, in 1980… T – When do they reach the height of 176.3 cm? In 1980 or in 1998? J – They reach 176.3 cm in 1980. T – Read the problem statement. Joana is a little bit confused and has difficulty in approaching the problem. The teacher scaffolds her thought process by asking questions. Specifically, the teacher suggests her to read the problem statement.**Task II - Lighthouse**2.3. Is there light or dark between the 30th and the 31st seconds? Justify Item type: Close response Competency cluster: Connections Overarching idea: Change and relationships Situation: Public**Strategy 1**Diana Reads the diagram and understands the period D-Ahhh. There is dark! T-Why do you say so? D-Because looking at the graph from the beginning, when I arrive here, there is dark. [Points to the time interval between 5 and 6 seconds]. T-Then between the 30th e 31st second what do you expect to happen? If you extrapolate from the diagram… D-If I extrapolate from the diagram to reach the time interval of the 30th to the 31st seconds, there will be dark. T-Why? How does that happen? D-Because the sequence is 2 seconds of dark followed by 1 second of light, followed by 1 second of dark and 1 second of light, and then this pattern is repeated throughout the graph. Diana analyses the sequence represented in the diagram and notes that it is repeated every 5 seconds. She further observes that between 30th and 31st seconds there is dark.**Strategy 2**Counting the number of light pulses in a sequence. R – Between the 30th and 31st seconds, is there light or dark? Explain. Then, every 5 periods there are light pulses… There is light and dark. Throughout the 60 seconds there is… T – We don´t want the 60. R – Ok, but throughout the 60 seconds, there are 24 light pulses. If throughout the 60 there are 24, throughout 30 we divide it by 2, 24 divided by 2 is 12. Then, throughout 30 seconds there will be 12 light pulses. T – I do not want to know how many light pulses there are. What I want to know is if between the 30th and 31st seconds the lighthouse is emitting a light pulse or not. Is the light shining or is it not? R – It is shining. T – How do you know that it is shining? R – Because it takes five seconds for the full sequence to occur… Raquel She counts the number of light pulses, associating it to a pattern (period).**Difficulties**Difficulty in the interpretation of the graphical representation (8th grade) • Light. Because I was counting with the same pattern and when it came to the 30th and 31st there was light. • Because following two seconds of dark there is one second of light • There is as much light as dark. • Because on every second the pattern shows that there is dark. • Because dark only shows up in the even numbers while light shows up in the odd numbers. • Yes. Because the lighthouse has two lights. (9th grade) • 30 dark 31 light. • There is light because at the end of 30 seconds there is light and it turns into dark.**Comments**Growing up • The quantitative data show that the performance of pupils involved in this study was slightly below both the national average, and the OCDE average. Lighthouse • The biggest problem encountered by pupils in this question was in understanding the symbolic language used and in recognizing and applying a pattern, • The quantitative data show that pupil performance is low - only 34% of the pupils involved in the study answered the question correctly.**Final reflection**In several questions we note different strategies from pupils, The interpretation issue – some pupils have trouble in understanding the information, When asked a question the pupil does not respond, but with other questions the pupil provides a correct answer, How to transform “partial” understanding in “full” understanding? Difficulty interpreting visual representation, Importance of how clearly the problem statement is written, in regards to both the language and symbols used.**Reasoning strategies and difficulties in visualization and**area Alexandra Simões, Ana Isabel Silvestre, Cristina Garcia, Guida Rocha, Mª José Molarinho, Nuno Candeias, Sandra Marques, Sara Costa**Theoretical framework - geometry**• The teaching of geometry • must be based on experimentation and manipulation and emphasize spatial visualization (ME, 2001; NCTM 2000). • Visual thinking • can be developed by composing and decomposing figures, along with its description of representation and reasoning about what happens (Abrantes, Serrazina & Oliveira, 1999). • Proportional reasoning • is one form of mathematical reasoning, involving a sense of covariation, multiple comparisons, and the ability to remember and process several pieces of information (Post, Behr & Lesh, 1988).**Methodology**• 80 pupils (10-11 years) • 70 written tests (45-60 min) • 10 interviews • Test and interview protocol with the same 6 questions.**Building blocks (Question 1.b)**How many small cubes will Susan need to make the solid block shown in Diagram C? • From written tests • 86% correct answers • Two strategies: • Calculate volume with formula, • Counting cube by cube.**Pupils’ strategies and difficulties**• (From interviews) • Because length is 3 and the width is 3, 3 times 3… 9 and the height is 3… 9 times 3 is… no… yes… 9 times 3 is 27. • I counted 9 cubes in the front that gives 27 because it has 9 blocks in front, in the middle and in the back.**Building blocks (Question 1.c)**Susan realizes that she used more small cubes than she really needed to make a block like the one shown in Diagram C. She realizes that she could have glued small cubes together to look like Diagram C, but the block could be hollow inside. What is the minimum number of cubes she needs to make a block like the one shown in Diagram C, but is hollow? Write down your reasoning.**Pupils’ strategies and difficulties**• (From written tests) • 56% correct answers. • (From interviews) • Difficulties • What is hollow? • Strategies • Subtracting one to 1.b answer, • Counting cube by cube.**Examples of pupils’ answers**(From written tests) (From interviews) “The minimum number of cubes is 24. I multiplied the cubes above by 3.” The pupil numbered all the cubes except the one in the middle and them multiplied by the other rows.**Comments**• Reflections about 1.b e 1.c • Observing 3D models decreases pupils’ difficulties, • Difficulty in understanding common words like “hollow”, • Information about pupils’ strategies is more evident in the interviews, • Using counting as a strategy is useful to begin formalizing the situation but the strategy based in the notion of volume is more efficient to give a correct answer to item 1.c.**Area of a continent (Question 3. a)**Find the distance between the South Pole and Mount Menzies. Write down your computations and reasoning. • (From written tests) • 27% correct answers.**Pupils’ strategies and difficulties**• (From interviews) • Strategies • Pupils used a ruler to measure the distance and then calculate using the scale of the map. • Difficulties • Working with decimal values, • Reading the scale of the map and converting the measurement.**Examples of pupils’ answers**(From interviews) P: I’m going to measure the distance. (Uses a ruler) T: And? P: Now I look to the map [points to the scale of the map]. It gives 850. T: Between 800 and 1000 km is…? P: Hum… Ah yes… is 900. The answer is 900 km. The pupil reads the scale as 1 cm corresponding to 100 km and not as 0.7 cm corresponding to 200 km.**Comments**• Question 3.a involves • Interpreting the situation, • Using a non usual scale in the map (0.7 cm 200 km), • Calculating with decimal numbers, • Using proportional reasoning. • Is 25% correct answers a good or a bad result?**Final reflection**Need for selecting better items • PISA items may be very good for PISA, but we need better items to understand pupils’ reasoning strategies and difficulties. Need for a more detailed interview protocol • A set of items is not enough as an interview guide – we need directions about styles of questioning and follow-up prompts. Need for theoretical foundations • PISA and PISA modified competences are not enough as a basis to interpret pupils thinking – we need topic specific theories. Need to provide enough time to data analysis and reflection on implications • Importance of “external presentation” fora.**Mathematics competence**PISA Competencies Framework PISA Modified • Representation and use of aids and tools • Thinking and reasoning • Including translating between representations and interpreting • Including using symbolic, formal and technical language and operations • Problem solving, Modelling, Investigating • Thinking and reasoning • Argumentation • Communication • Modelling • Problem posing and solving • Representation • Using symbolic, formal and technical language and operations • Use of aids and tools 4. Communication and Argumentation including proving Always in two contexts: Mathematics and real life**PDTR Collaboration**• Preparing • Collecting • Analysing

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