

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, welldeveloped 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 realworld situations.
Students at this level can utilise welldeveloped 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 problemsolving 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.
