An Introduction to the PISA Maths Assessment

MATHEMATICS

An introduction to the PISA maths assessment Are students prepared to meet the challenges of the future? Are they able to analyse, reason and communicate their ideas effectively? How well equipped are they to continue learning throughout life?

The OECD Programme for International Student Assessment (PISA) aims to answer these questions through three-yearly surveys that examine 15-year-old students’ performance in reading, mathematics and science. The first four surveys addressed these subjects in 2000, 2003, 2006 and 2012 respectively. PISA 2012 will focus on mathematics as well assessing performance in science and reading.

PISA focuses on testing the knowledge and skills required for participation in society and assessing the extent to which students can apply skills gained in school in everyday adult life, thus moving beyond the student’s ability to master the school curriculum.

Worldwide, about 470,000 students from 65 countries took part in the study in 2009.

Mathematics assessment

The mathematics assessment seeks to measure the student’s capacity to analyse, reason and communicate effectively as they pose, solve and interpret mathematical problems in a variety of situations that involve quantity, space, probability and other mathematical concepts. In particular, the maths assessment focuses on the following areas and concepts:

• Quantity • Space and shape • Change and relationships • Uncertainty

On the following pages are a selection of questions that show what can be expected during the maths assessment. Questions in PISA cover a wide range of topics and levels of difficulty in order to allow the survey to measure the range of abilities of students as well as to draw comparisons internationally. Details of the level of difficulty and what proportion of students answered the question correctly are included for reference. The tasks are taken from PISA items released by the OECD for public use. Copyright remains with the OECD. Levels of difficulty as defined by PISA and the assessment of student’s ability LEVEL What students can typically do At Level 6 students can conceptualise, generalise and utilise information based on their investigations and modelling of complex problem situations. They can link different information sources and representations and flexibly translate between them. Students at this level are capable of advanced mathematical thinking and reasoning. These students 6 can apply this insight and understanding along with a mastery of symbolic and formal mathematical operations and relationships to develop new approaches and strategies for attacking novel situations. Students at this level can formulate and precisely communicate their actions and reflections regarding their findings, interpretations, arguments, and the appropriateness of these to the original situations. At Level 5 students can develop and work with models for complex situations, identifying constraints and specifying assumptions. They can select, compare, and evaluate appropriate problem-solving strategies for dealing with complex problems related to these 5 models. Students at this level can work strategically using broad, well-developed thinking and reasoning skills, appropriately linked representations, symbolic and formal characterisations, and insight pertaining to these situations. They can reflect on their actions and formulate and communicate their interpretations and reasoning. At Level 4 students can work effectively with explicit models for complex concrete situations that may involve constraints or call for making assumptions. They can select and integrate different representations, including symbolic representations, linking them directly 4 to aspects of real-world situations. Students at this level can utilise well-developed skills and reason flexibly, with some insight, in these contexts. They can construct and communicate explanations and arguments based on their interpretations, arguments and actions. At Level 3 students can execute clearly described procedures, including those that require sequential decisions. They can select and apply simple problem-solving strategies. 3 Students at this level can interpret and use representations based on different information sources and reason directly from them. They can develop short communications reporting their interpretations, results and reasoning. At Level 2 students can interpret and recognise situations in contexts that require no more than direct inference. They can extract relevant information from a single source and make 2 use of a single representational mode. Students at this level can employ basic algorithms, formulae, procedures, or conventions. They are capable of direct reasoning and literal interpretations of the results. At Level 1 students can answer questions involving familiar contexts where all relevant information is present and the questions are clearly defined. They are able to identify 1 information and to carry out routine procedures according to direct instructions in explicit situations. They can perform actions that are obvious and follow immediately from the given stimuli. STAIRCASE

The diagram below illustrates a staircase with 14 steps and a total height of 252 cm:

What is the height of each of the 14 steps?

Height: ...... cm.

Scoring

Full Credit: 18 Percentage of correct answers (OECD countries): 78.3%

Comments: This short open-constructed response item is situated in a daily life context for carpenters and is therefore classified as having an occupational context. One does not need to be a carpenter to understand the relevant information; it is clear that an informed citizen should be able to interpret and solve a problem like this that uses two different representation modes: language, including numbers, and a graphical representation. But the illustration serves a simple and non-essential function: students know what stairs look like. This item is noteworthy because it has redundant information (the depth is 400cm) that is sometimes considered to be confusing by students; but such redundancy is common in real-world problem solving. The context of the stairs places the item in the space and shape content area, but the actual procedure to carry out is simple division. All the required information, and even more than that, is presented in a recognisable situation, and the students can extract the relevant information from a single source. In essence, the item makes use of a single representational mode, and with the application of a basic algorithm, this item fits, although barely, at Level 2. CARPENTER

A carpenter has 32 metres of timber and wants to make a border around a garden bed. He is considering the following designs for the garden bed.

Circle either “Yes” or “No” for each design to indicate whether the garden bed can be made with 32 metres of timber. Scoring

Full Credit: Yes, No, Yes, Yes, in that order. Percentage of correct answers (OECD countries): 20.2%

Comments This complex multiple-choice item is situated in an educational context, since it is the kind of quasi-realistic problem that would typically be seen in a mathematics class, rather than being a genuine problem likely to be met in an occupational setting. A small number of such problems have been included in PISA, though they are not typical. That being said, the competencies needed for this problem are certainly relevant and part of mathematical literacy. The item belongs to the space and shape content area. The students need the competence to recognise that the two- dimensional shapes A, C and D have the same perimeter, and therefore they need to decode the visual information and see similarities and differences. The students need to see whether or not a certain border-shape can be made with 32 metres of timber. In three cases this is rather evident because of the rectangular shapes. But the fourth is a parallelogram, requiring more than 32 metres. This use of geometrical insight, argumentation skills and some technical geometrical knowledge puts this item at Level 6. EXCHANGE RATE

Mei-Ling from Singapore was preparing to go to South Africa for 3 months as an exchange student. She needed to change some Singapore dollars (SGD) into South African rand (ZAR).

Question 1: EXCHANGE RATE

Mei-Ling found out that the exchange rate between Singapore dollars and South African rand was:

1 SGD = 4.2 ZAR

Mei-Ling changed 3000 Singapore dollars into South African rand at this exchange rate.

How much money in South African rand did Mei-Ling get?

Answer: ......

Scoring

Full Credit: 12 600 ZAR (unit not required). Percentage of correct answers (OECD countries): 79.9%

Comments This short open-constructed response item is situated in a public context. Experience in using exchange rates may not be common to all students, but the concept can be seen as belonging to skills and knowledge for citizenship. The mathematics content is restricted to just one of the four basic operations: multiplication. This places the item in the quantity area, and more specifically, in operations with numbers. As far as the competencies are concerned, a very limited form of mathematisation is needed for understanding a simple text and linking the given information to the required calculation. All the required information is explicitly presented. Thus the competency needed to solve this problem can be described as the performance of a routine procedure and/or application of a standard algorithm. The combination of a familiar context, a clearly defined question and a routine procedure places the item at Level 1. Question 2: EXCHANGE RATE

During these 3 months the exchange rate had changed from 4.2 to 4.0 ZAR per SGD.

Was it in Mei-Ling’s favour that the exchange rate now was 4.0 ZAR instead of 4.2 ZAR, when she changed her South African rand back to Singapore dollars? Give an explanation to support your answer.

Answer: ......

Scoring

Full Credit: Yes, with adequate explanation. Percentage of correct answers (OECD countries): 40.5%

Comments This open-constructed response item is situated in a public context. As far as the mathematics content is concerned students need to apply procedural knowledge involving number operations: multiplication and division, which along with the quantitative context, place the item in the quantity area. The competencies needed to solve the problem are not trivial. Students need to reflect on the concept of exchange rate and its consequences in this particular situation. The mathematisation required is of a rather high level, although all the required information is explicitly presented: not only is the identification of the relevant mathematics somewhat complex, but the reduction of it to a problem within the mathematical world also places significant demands on the student. The competency needed to solve this problem can be described as using flexible reasoning and reflection. Explaining the results requires some communication skills as well. The combination of familiar context, complex situation, non-routine problem and the need for reasoning, insight and communication places the item at Level 4. TEST SCORES

The diagram below shows the results on a Science test for two groups, labelled as Group A and Group B. The mean score for Group A is 62.0 and the mean for Group B is 64.5. Students pass this test when their score is 50 or above.

Looking at the diagram, the teacher claims that Group B did better than Group A in this test. The students in Group A don’t agree with their teacher. They try to convince the teacher that Group B may not necessarily have done better.

Give one mathematical argument, using the graph, that the students in Group A could use.

Answer: ...... Scoring

Full credit: One valid argument is given. Valid arguments could relate to the number of students passing, the disproportionate influence of the outlier, or the number of students with scores in the highest level.

• More students in Group A than in Group B passed the test. • If you ignore the weakest Group A student, the students in Group A do better than those in Group B. • More Group A students than Group B students scored 80 or over.

Percentage of correct answers (OECD countries): 32.7%

Comments This open-constructed response item is situated in an educational context. The educational context of this item is one that all students are familiar with: comparing test scores. In this case a science test has been administered to two groups of students: A and B. The results are given to the students in two different ways: in words with some data embedded and by means of two graphs in one grid. Students must find arguments that support the statement that Group A actually did better than Group B, given the counter-argument of one teacher that Group B did better – on the grounds of the higher mean for Group B. The item falls into the content area of uncertainty. Knowledge of this area of mathematics is essential, as data and graphical representations play a major role in the media and in other aspects of daily experiences. The students have a choice of at least three arguments here: the first one is that more students in Group A pass the test; a second one is the distorting effect of the outlier in the results of Group A; and a final argument is that Group A has more students that scored 80 or above. Students who are successful have applied statistical knowledge in a problem situation that is somewhat structured and where the mathematical representation is partially apparent. They need reasoning and insight to interpret and analyse the given information, and they must communicate their reasons and arguments. Therefore the item clearly illustrates Level 5. Here are a few more past PISA numeracy questions.

LITTER For a homework assignment on the environment, students collected information on the decomposition time of several types of litter that people throw away:

Type of Litter Decomposition time

Banana peel 1–3 years

Orange peel 1–3 years

Cardboard boxes 0.5 year

Chewing gum 20–25 years

Newspapers A few days

Polystyrene cups Over 100 years

Question 1: LITTER A student thinks of displaying the results in a bar graph. Give one reason why a bar graph is unsuitable for displaying these data.

EARTHQUAKE

Question 2: EARTHQUAKE A documentary was broadcast about earthquakes and how often earthquakes occur. It included a discussion about the predictability of earthquakes. A geologist stated: “In the next twenty years, the chance that an earthquake will occur in Zed City is two out of three”.

Which of the following best reflects the meaning of the geologist’s statement?

A 2/3 x 20 = 13.3, so between 13 and 14 years from now there will be an earthquake in Zed City. B 2/3 is more than ½, so you can be sure there will be an earthquake in Zed City at some time during the next 20 years. C The likelihood that there will be an earthquake in Zed City at some time during the next 20 years is higher than the likelihood of no earthquake. D You cannot tell what will happen, because nobody can be sure when an earthquake will occur. GROWING UP

YOUTH GROWS TALLER In 1998 the average height of both young males and young females in the Netherlands is represented in this graph.

Height (cm) 190 Average height of young males 1998 180

Average height of 170 young females 1998

10 11 12 13 14 15 16 17 18 19 20 Age (years)

Question 3: GROWING UP 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?

Answer: ...... cm

Question 4: GROWING UP Explain how the graph shows that on average the growth rate for girls slows down after 12 years of age......

......

...... Question 5: GROWING UP According to this graph, on average, during which period in their life are females taller than males of the same age? ......

......

LICHEN A result of global warming is that the ice of some glaciers is melting. Twelve years after the ice disappears, tiny plants, called lichen, start to grow on the rocks. Each lichen grows approximately in the shape of a circle.

The relationship between the diameter of this circle and the age of the lichen can be approximated with the formula: d =7.0×√(t − 12) for t ≥ 12 where d represents the diameter of the lichen in millimetres, and t represents the number of years after the ice has disappeared.

Question 6: LICHEN Using the formula, calculate the diameter of the lichen, 16 years after the ice disappeared. Show your calculation.

Question 7: LICHEN Ann measured the diameter of some lichen and found it was 35 millimetres. How many years ago did the ice disappear at this spot? Show your calculation. LIGHTHOUSE Lighthouses are towers with a light beacon on top. Lighthouses assist sea ships in finding their way at night when they are sailing close to the shore.

A lighthouse beacon sends out light flashes with a regular fixed pattern. Every lighthouse has its own pattern.

In the diagram below you see the pattern of a certain lighthouse. The light flashes alternate with dark periods. light dark

0 1 2 3 4 5 6 7 8 9 10 11 12 13 time (sec)

It is a regular pattern. After some time the pattern repeats itself. The time taken by one complete cycle of a pattern, before it starts to repeat, is called the period. When you find the period of a pattern, it is easy to extend the diagram for the next seconds or minutes or even hours.

Question 8: LIGHTHOUSE Which of the following could be the period of the pattern of this lighthouse? A 2 seconds. B 3 seconds. C 5 seconds. D 12 seconds.

Question 9: LIGHTHOUSE For how many seconds does the lighthouse send out light flashes in 1 minute? A 4 B 12 C 20 D 24

Question 10: LIGHTHOUSE In the diagram below, make a graph of a possible pattern of light flashes of a lighthouse that sends out light flashes for 30 seconds per minute. The period of this pattern must be equal to 6 seconds. light dark

0 1 2 3 4 5 6 7 8 9 10 11 12 time (sec) BUILDING BLOCKS Susan likes to build blocks from small cubes like the one shown in the following diagram:

Small cube

Susan has lots of small cubes like this one. She uses glue to join cubes together to make other blocks.

First, Susan glues eight of the cubes together to make the block shown in Diagram A:

Diagram A

Then Susan makes the solid blocks shown in Diagram B and Diagram C below:

Diagram B Diagram C

Question 11: BUILDING BLOCKS How many small cubes will Susan need to make the block shown in Diagram B?

Answer: ...... cubes.

Question 12: BUILDING BLOCKS How many small cubes will Susan need to make the solid block shown in Diagram C?

Answer: ...... cubes. Question 13: BUILDING BLOCKS Susan realises that she used more small cubes than she really needed to make a block like the one shown in Diagram C. She realises that she could have glued small cubes together to look like Diagram C, but the block could have been hollow on the inside. What is the minimum number of cubes she needs to make a block that looks like the one shown in Diagram C, but is hollow?

Answer: ...... cubes.

Question 14: BUILDING BLOCKS Now Susan wants to make a block that looks like a solid block that is 6 small cubes long, 5 small cubes wide and 4 small cubes high. She wants to use the smallest number of cubes possible, by leaving the largest possible hollow space inside the block. What is the minimum number of cubes Susan will need to make this block?

Answer: ...... cubes. TABLE TENNIS TOURNAMENT

Question 15: TABLE TENNIS TOURNAMENT Teun, Riek, Bep and Dirk have formed a practice group in a table tennis club. Each player wishes to play against each other player once. They have reserved two practice tables for these matches. Complete the following match schedule; by writing the names of the players playing in each match.

Practice Table 1 Practice Table 2

Round 1 Teun - Riek Bep - Dirk

Round 2 ……………-………….. ……………-………….. Round 3 ……………-………….. ……………-………….. COLOURED SWEETS Question 16: Robert’s mother lets him pick one sweet from a bag. He can’t see the sweets. The number of sweets of each colour in the bag is shown in the following graph.

0 k d e n e e n w l n e u g e o w i l p l n e r l R o P r B r a e u r G B Y P O

What is the probability that Robert will pick a red sweet? A 10% B 20% C 25% D 50%

Question 17: CUBES In this photograph you see six dice, labelled (a) to (f). For all dice there is a rule: The total number of dots on two opposite faces of each die is always seven.

Write in each box the number of dots on the bottom face of the dice corresponding to the photograph. (a) (b) (c)

(d) (e) (f)

WALKING

The picture shows the footprints of a man walking. The pacelength P is the distance between the rear of two consecutive footprints. For men, the formula, n / P =140 gives an approximate relationship between n and P where, n = number of steps per minute, and P = pacelength in metres.

Question 18: WALKING If the formula applies to Heiko’s walking and Heiko takes 70 steps per minute, what is Heiko’s pacelength? Show your work.

Question 19: WALKING Bernard knows his pacelength is 0.80 metres. The formula applies to Bernard’s walking. Calculate Bernard’s walking speed in metres per minute and in kilometres per hour. Show your working out. THE BEST CAR A car magazine uses a rating system to evaluate new cars, and gives the award of “The Car of the Year” to the car with the highest total score. Five new cars are being evaluated, and their ratings are shown in the table.

Car Safety Fuel External Internal Features Efficiency Appearance Fittings

(S) (F) (E) (T) Ca 3 1 2 3

M2 2 2 2 2

Sp 3 1 3 2

N1 1 3 3 3

KK 3 2 3 2

The ratings are interpreted as follows: 3 points = Excellent 2 points = Good 1 point = Fair

Question 20: THE BEST CAR To calculate the total score for a car, the car magazine uses the following rule, which is a weighted sum of the individual score points:

Total Score = (3 x S) + F + E + T

Calculate the total score for Car “Ca”. Write your answer in the space below.

Total score for “Ca”: ......

Question 21: THE BEST CAR The manufacturer of car “Ca” thought the rule for the total score was unfair. Write down a rule for calculating the total score so that Car “Ca” will be the winner. Your rule should include all four of the variables, and you should write down your rule by filling in positive numbers in the four spaces in the equation below.

Total score = ………× S + ………× F + ………× E + ………× T. MOVING WALKWAYS

Question 22: MOVING WALKWAYS On the right is a photograph of moving walkways.

The following Distance-Time graph shows a comparison between “walking on the moving walkway” and “walking on the ground next to the moving walkway.”

Distance from the start of the moving walkway A person walking on the moving walkway

A person walking on the ground

time Assuming that, in the above graph, the walking pace is about the same for both persons, add a line to the graph that would represent the distance versus time for a person who is standing still on the moving walkway. TWISTED BUILDING In modern architecture, buildings often have unusual shapes. The picture below shows a computer model of a ‘twisted building’ and a plan of the ground floor. The compass points show the orientation of the building.

The ground floor of the building contains the main entrance and has room for shops. Above the ground floor there are 20 storeys containing apartments. The plan of each storey is similar to the plan of the ground floor, but each has a slightly different orientation from the storey below. The cylinder contains the elevator shaft and a landing on each floor.

Question 23: TWISTED BUILDING Estimate the total height of the building, in metres. Explain how you found your answer. The following pictures are sideviews of the twisted building.

sideview 1 sideview 2

Question 24: TWISTED BUILDING From which direction has Sideview 1 been drawn?

A From the North. B From the West. C From the East. D From the South.

Question 25: TWISTED BUILDING From which direction has Sideview 2 been drawn?

A From the North West. B From the North East. C From the South West. D From the South East. Question 26: TWISTED BUILDING Each storey containing apartments has a certain ‘twist’ compared to the ground floor. The th top floor (the 20 floor above the ground floor) is at right angles to the ground floor. The drawing below represents the ground floor.

th Draw in this diagram the plan of the 10 floor above the ground floor, showing how this floor is situated compared to the ground floor. For further information about PISA in Scotland please contact Mal Cooke on 0131 244 1689 or [email protected] www.scotland.gov.uk/pisa