Mathematics literacy achievement: senior secondary schooling
Mathematical attainment at senior secondary level contributes to preparation for successful participation in tertiary education, and the ability to contribute to, and participate in, a changing labour market and an increasingly knowledge-based society. Attainment level is also related to individual well being.
Mathematical attainment is also important because mathematical literacy is the ability to formulate and solve mathematical problems in real life situations. This type of literacy is a foundation for participation as a reflective citizen in democracy and in occupational life.
Methodology behind the indicator
The mathematics scores from the Programme for International Student Assessment (PISA) 2003, 2006 2009, 2012 and 2015 study years can be summarised on a combined mathematical literacy scale. This enables a comparison to be made between the mathematics literacy achievements of 15 year-old students in each of these years.
Because the mathematical literacy domain underwent considerable expansion and change between 2000 and 2003, mathematical outcomes from PISA 2003 onwards are not comparable with results from PISA 2000.
The Item Response Theory (IRT) scaling approach and plausible values methodology is used in PISA. This involved estimating the parameters for each item (question) and examining the background characteristics of the students. From this, estimates of proficiency for each student and IRT scales for reporting student achievement were generated; in aggregate and for each major content domain. Finally, the resulting values were placed on a reporting scale in PISA 2003 with a mean of 500 and standard deviation of 100. Subsequent cycles (2006, 2009, 2012 and 2015) were anchored against the PISA 2003 scale. This enables a comparison to be made between the mathematical literacy achievement of 15 year-old students in each of 2003, 2006, 2009, 2012 and 2015. The IRT analysis provided a common scale on which the performances of students within and across countries may be compared.
In 2015 each student has 10 estimates of ability called plausible values (PV1-PV10). From 2003 to 2012 each student had 5 plausible values (PV1-PV5). For each student, the plausible values represent a set of random draws from the estimated ability distribution of students with similar item response patterns and backgrounds. They are intended to provide good estimates of parameters of student populations, for example, country mean scores, rather than estimates of individual student proficiency.
For any group of 15 year-old students, for example, the New Zealand Population, Māori, or Girls, the numerator and denominator are defined as follows:
Sum of the mean mathematics literacy scores for each plausible value for that group.
[Where the mean for each plausible value is defined as:
- Numerator: Weighted sum of scores for that group.
- Denominator: Sum of the weights for that group (equivalent to the estimated number of students in that group).]
(Data Source: OECD Programme for International Student Assessment (PISA))
n (number of plausible values).
(Data Source: OECD Programme for International Student Assessment (PISA))
Mean PISA scores for the New Zealand population and sub-populations are based on scores generated using Item Response Theory. These scores are reported on an international scale with an international mean of 500 and a standard deviation of 100 for OECD countries so that approximately two-thirds of all students internationally have a score between 400 and 600.
In PISA 2015 proficiency levels related to the difficulty of the tasks that students were assessed on, with each content area having its own set of proficiency levels. These range from Level 1 for the simplest tasks to Level 6 for the most complex. For information on the proficiency levels for each content area see: OECD (2016). PISA 2015 Results: Excellence and Equity in Education (Volume I). OECD: Paris.
|Level||Lower score limit|
Characteristics of Tasks
|6||669||At Level 6, students can conceptualise, generalise and utilise information based on their investigations and modelling of complex problem situations, and can use their knowledge in relatively non-standard contexts. They can link different information sources and representations and flexibly translate among them. Students at this level are capable of advanced mathematical thinking and reasoning. These students 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 reflect on their actions, and can formulate and precisely communicate their actions and reflections regarding their findings, interpretations, arguments, and the appropriateness of these to the original situation.|
|5||607||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 models. Students at this level can work strategically using broad, well- developed thinking and reasoning skills, appropriate linked representations, symbolic and formal characterisations, and insight pertaining to these situations. They begin to reflect on their work and can formulate and communicate their interpretations and reasoning.|
|4||545||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, linking them directly to aspects of real-world situations. Students at this level can utilise their limited range of skills and can reason with some insight, in straightforward contexts. They can construct and communicate explanations and arguments based on their interpretations, arguments and actions.|
|3||482||At Level 3, students can execute clearly described procedures, including those that require sequential decisions. Their interpretations are sufficiently sound to be a base for building a simple model or for selecting and applying simple problem-solving strategies. Students at this level can interpret and use representations based on different information sources and reason directly from them. They typically show some ability to handle percentages, fractions and decimal numbers, and to work with proportional relationships. Their solutions reflect that they have engaged in basic interpretation and reasoning.|
|2||420||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 use of a single representational mode. Students at this level can employ basic algorithms, formulae, procedures or conventions to solve problems involving whole numbers. They are capable of making literal interpretations of the results.|
|1||358||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 information and to carry out routine procedures according to direct instructions in explicit situations. They can perform actions that are almost always obvious and follow immediately from the given stimuli.|