## Mathematics proficiency levels

Mathematics achievement was divided into six proficiency levels representing a group of tasks of increasing difficulty, with Level 6 as the highest and Level 1 as the lowest. Students performing below Level 1 (mathematics score below 359) are not able to show routinely the most basic type of knowledge and skills that PISA seeks to measure. Such students have serious difficulties in using mathematical literacy as a tool to advance their knowledge and skills in other areas. Placement at this level does not mean that these students have no mathematics skills. Most of these students are able to correctly complete some of the PISA items. Their pattern of responses to the assessment is such that they would be expected to solve less than half of the tasks from a test composed of only Level 1 items.

In PISA, students were assigned to a proficiency level based on their probability of answering correctly themajority of items in that range of difficulty. A student at a given level could be assumed to be able to correctly answer questions at all lower levels. To help in interpretation, these levels were linked to specific score ranges on the original scale. Below is a description of the abilities associated with each proficiency level. (Source: Organisation for Economic Cooperation and Development, Programme for International Student Assessment, PISA, 2003).

Level 6 (score above 668)

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 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 formulate and precisely communicate their actions and reflections regarding their findings, interpretations, arguments, and the appropriateness of these to the original situations.

Level 5 (score from 607 to 668)

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 can reflect on their actions and formulate and communicate their interpretations and reasoning.

Level 4 (score from 545 to 606)

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 ones, linking them directly 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.

Level 3 (score from 483 to 544)

At Level 3 students can execute clearly described procedures, including those that require sequential decisions. They can select and apply simple problem-solving strategies. 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.

Level 2 (score from 421 to 482)

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. They are capable of direct reasoning and of making literal interpretations of the results.

Level 1 (score from 359 to 420)

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 obvious and follow immediately from the given stimuli.

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