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
Maria José Molarinho, Gaspar Correia School, Portela, Loures
July 2006
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)
Who is who
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
Objectives
To know the reasoning strategies and
dificulties of Portuguese pupils in
 Algebra: Patterns, Simbolization and
Change
Preparation
 Organizing target
populations (pupils
aged 11 and 13-14)
 Organizing teams
 Selecting instruments
 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
Qualitative and
interpretative
Written test with 15
questions (94 pupils)
Descriptive
and interpretative
General aspects
Data collected
Data analysis
101 pupils
13-15 years old
Interview with 10
questions (7 pupils)
Focus on
algebra
Task I – Growing up
In 1998 the average height of both young males and young
females in the Netherlands is represented in this graph.
190
Height
(cm)
Average height of young
males - 1998
180
Average height of young
females - 1998
170
160
150
140
130
10 11 12 13 14 15 16 17 18 19 20
1.2. Since 1980 the
average height of 20-yearold females has increased
by 2.3 cm, to 170.6 cm.
What was the average
height of a 20-year-old
female in 1980?
Age
(Years)
Item type:
Close-constructed response
Competency cluster:
Reproduction
Overarching idea:
Change and relationships
Situation:
Scientific
Strategy 1
The pupil reads the question and see that they can solve it using the
given information.
In 1980, 20-year-old girls measure 168.3 cm
Diana
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.
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.
Raquel
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…
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
1. Thinking and
reasoning
2. Argumentation
3. Communication
4. Modelling
5. Problem posing and
solving
6. Representation
7. Using symbolic,
formal and technical
language and
operations
8. Use of aids and tools
PISA Modified
1. Representation and use
of aids and tools
2. Thinking and reasoning
 Including translating
between representations
and interpreting
 Including using
symbolic, formal and
technical language and
operations
3. Problem solving,
Modelling, Investigating
4. Communication and Argumentation
including proving
PISA Competencies
Framework
Always in two contexts: Mathematics and real life
PDTR Collaboration
 Preparing
 Collecting
 Analysing