Order Description

Purpose of internal assessment
Internal assessment is an integral part of the course and is compulsory for all students. It enables students to demonstrate the application of their skills and knowledge, and to pursue their personal interests, without the time limitations and other constraints that are associated with written examinations. The internal assessment should, as far as possible, be woven into normal classroom teaching and not be a separate activity conducted after a course has been taught.
Internal assessment in mathematics HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. It is marked according to five assessment criteria.
Guidance and authenticity
The exploration submitted for internal assessment must be the student’s own work. However, it is not the intention that students should decide upon a title or topic and be left to work on the exploration without any further support from the teacher. The teacher should play an important role during both the planning stage and the period when the student is working on the exploration. It is the responsibility of the teacher to ensure that students are familiar with:
• the requirements of the type of work to be internally assessed
• the IB academic honesty policy available on the OCC
• the assessment criteria—students must understand that the work submitted for assessment must address these criteria effectively.
Teachers and students must discuss the exploration. Students should be encouraged to initiate discussions with the teacher to obtain advice and information, and students must not be penalized for seeking guidance. However, if a student could not have completed the exploration without substantial support from the teacher, this should be recorded on the appropriate form from the Handbook of procedures for the Diploma Programme.
It is the responsibility of teachers to ensure that all students understand the basic meaning and significance of concepts that relate to academic honesty, especially authenticity and intellectual property. Teachers must ensure that all student work for assessment is prepared according to the requirements and must explain clearly to students that the exploration must be entirely their own.
As part of the learning process, teachers can give advice to students on a first draft of the exploration. This advice should be in terms of the way the work could be improved, but this first draft must not be heavily annotated or edited by the teacher. The next version handed to the teacher after the first draft must be the final one.
All work submitted to the IB for moderation or assessment must be authenticated by a teacher, and must not include any known instances of suspected or confirmed malpractice. Each student must sign the coversheet for internal assessment to confirm that the work is his or her authentic work and constitutes the final version of that work. Once a student has officially submitted the final version of the work to a teacher (or the coordinator) for internal assessment, together with the signed coversheet, it cannot be retracted.
64 Mathematics HL guide

Authenticity may be checked by discussion with the student on the content of the work, and scrutiny of one or more of the following:
• the student’s initial proposal
• the first draft of the written work
• the references cited
• the style of writing compared with work known to be that of the student.
The requirement for teachers and students to sign the coversheet for internal assessment applies to the work of all students, not just the sample work that will be submitted to an examiner for the purpose of moderation. If the teacher and student sign a coversheet, but there is a comment to the effect that the work may not be authentic, the student will not be eligible for a mark in that component and no grade will be awarded. For further details refer to the IB publication Academic honesty and the relevant articles in the General regulations: Diploma Programme.
The same piece of work cannot be submitted to meet the requirements of both the internal assessment and the extended essay.
Group work
Group work should not be used for explorations. Each exploration is an individual piece of work based on different data collected or measurements generated.
It should be made clear to students that all work connected with the exploration, including the writing of the exploration, should be their own. It is therefore helpful if teachers try to encourage in students a sense of responsibility for their own learning so that they accept a degree of ownership and take pride in their own work.
Time allocation
Internal assessment is an integral part of the mathematics HL course, contributing 20% to the final assessment in the course. This weighting should be reflected in the time that is allocated to teaching the knowledge, skills and understanding required to undertake the work as well as the total time allocated to carry out the work.
It is expected that a total of approximately 10 teaching hours should be allocated to the work. This should include:
• time for the teacher to explain to students the requirements of the exploration
• class time for students to work on the exploration
• time for consultation between the teacher and each student
• time to review and monitor progress, and to check authenticity.
Using assessment criteria for internal assessment
For internal assessment, a number of assessment criteria have been identified. Each assessment criterion has level descriptors describing specific levels of achievement together with an appropriate range of marks. The level descriptors concentrate on positive achievement, although for the lower levels failure to achieve may be included in the description.
Mathematics HL guide 65
Internal assessment

Internal assessment
Teachers must judge the internally assessed work against the criteria using the level descriptors.
• The aim is to find, for each criterion, the descriptor that conveys most accurately the level attained by the student.
• When assessing a student’s work, teachers should read the level descriptors for each criterion, starting with level 0, until they reach a descriptor that describes a level of achievement that has not been reached. The level of achievement gained by the student is therefore the preceding one, and it is this that should be recorded.
• Only whole numbers should be recorded; partial marks, that is fractions and decimals, are not acceptable.
• Teachers should not think in terms of a pass or fail boundary, but should concentrate on identifying the appropriate descriptor for each assessment criterion.
• The highest level descriptors do not imply faultless performance but should be achievable by a student. Teachers should not hesitate to use the extremes if they are appropriate descriptions of the work being assessed.
• A student who attains a high level of achievement in relation to one criterion will not necessarily attain high levels of achievement in relation to the other criteria. Similarly, a student who attains a low level of achievement for one criterion will not necessarily attain low achievement levels for the other criteria. Teachers should not assume that the overall assessment of the students will produce any particular distribution of marks.
• It is expected that the assessment criteria be made available to students.
Internal assessment details
Mathematical exploration
Duration: 10 teaching hours Weighting: 20%
Introduction
The internally assessed component in this course is a mathematical exploration. This is a short report written by the student based on a topic chosen by him or her, and it should focus on the mathematics of that particular area. The emphasis is on mathematical communication (including formulae, diagrams, graphs and so on), with accompanying commentary, good mathematical writing and thoughtful reflection. A student should develop his or her own focus, with the teacher providing feedback via, for example, discussion and interview. This will allow the students to develop areas of interest to them without a time constraint as in an examination, and allow all students to experience a feeling of success.
The final report should be approximately 6 to 12 pages long. It can be either word processed or handwritten. Students should be able to explain all stages of their work in such a way that demonstrates clear understanding. While there is no requirement that students present their work in class, it should be written in such a way that their peers would be able to follow it fairly easily. The report should include a detailed bibliography, and sources need to be referenced in line with the IB academic honesty policy. Direct quotes must be acknowledged.
The purpose of the exploration
The aims of the mathematics HL course are carried through into the objectives that are formally assessed as part of the course, through either written examination papers, or the exploration, or both. In addition to testing the objectives of the course, the exploration is intended to provide students with opportunities to increase their understanding of mathematical concepts and processes, and to develop a wider appreciation of mathematics. These are noted in the aims of the course, in particular, aims 6–9 (applications, technology, moral, social
66 Mathematics HL guide

and ethical implications, and the international dimension). It is intended that, by doing the exploration, students benefit from the mathematical activities undertaken and find them both stimulating and rewarding. It will enable students to acquire the attributes of the IB learner profile.
The specific purposes of the exploration are to:
• develop students’ personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics
• provide opportunities for students to complete a piece of mathematical work over an extended period of time
• enable students to experience the satisfaction of applying mathematical processes independently
• provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics
• encourage students, where appropriate, to discover, use and appreciate the power of technology as a mathematical tool
• enable students to develop the qualities of patience and persistence, and to reflect on the significance of their work
• provide opportunities for students to show, with confidence, how they have developed mathematically.
Management of the exploration
Work for the exploration should be incorporated into the course so that students are given the opportunity to learn the skills needed. Time in class can therefore be used for general discussion of areas of study, as well as familiarizing students with the criteria. Further details on the development of the exploration are included in the teacher support material.
Requirements and recommendations
Students can choose from a wide variety of activities, for example, modelling, investigations and applications of mathematics. To assist teachers and students in the choice of a topic, a list of stimuli is available in the teacher support material. However, students are not restricted to this list.
The exploration should not normally exceed 12 pages, including diagrams and graphs, but excluding the bibliography. However, it is the quality of the mathematical writing that is important, not the length.
The teacher is expected to give appropriate guidance at all stages of the exploration by, for example, directing students into more productive routes of inquiry, making suggestions for suitable sources of information, and providing advice on the content and clarity of the exploration in the writing-up stage.
Teachers are responsible for indicating to students the existence of errors but should not explicitly correct these errors. It must be emphasized that students are expected to consult the teacher throughout the process.
All students should be familiar with the requirements of the exploration and the criteria by which it is assessed. Students need to start planning their explorations as early as possible in the course. Deadlines should be firmly established. There should be a date for submission of the exploration topic and a brief outline description, a date for the submission of the first draft and, of course, a date for completion.
In developing their explorations, students should aim to make use of mathematics learned as part of the course. The mathematics used should be commensurate with the level of the course, that is, it should be similar to that suggested by the syllabus. It is not expected that students produce work that is outside the mathematics HL syllabus—however, this is not penalized.
Mathematics HL guide 67
Internal assessment

Internal assessment
Internal assessment criteria
The exploration is internally assessed by the teacher and externally moderated by the IB using assessment criteria that relate to the objectives for mathematics HL.
Each exploration is assessed against the following five criteria. The final mark for each exploration is the sum of the scores for each criterion. The maximum possible final mark is 20.
Students will not receive a grade for mathematics HL if they have not submitted an exploration.
Criterion A
Communication
Criterion B
Mathematical presentation
Criterion C
Personal engagement
Criterion D
Ref lection
Criterion E
Use of mathematics
Criterion A: Communication
This criterion assesses the organization and coherence of the exploration. A well-organized exploration includes an introduction, has a rationale (which includes explaining why this topic was chosen), describes the aim of the exploration and has a conclusion. A coherent exploration is logically developed and easy to follow.
Graphs, tables and diagrams should accompany the work in the appropriate place and not be attached as appendices to the document.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
The exploration has some coherence.
2
The exploration has some coherence and shows some organization.
3
The exploration is coherent and well organized.
4
The exploration is coherent, well organized, concise and complete.
Criterion B: Mathematical presentation
This criterion assesses to what extent the student is able to:
• use appropriate mathematical language (notation, symbols, terminology)
• define key terms, where required
• use multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs and models, where appropriate.
Students are expected to use mathematical language when communicating mathematical ideas, reasoning and findings.
68 Mathematics HL guide

Internal assessment
Students are encouraged to choose and use appropriate ICT tools such as graphic display calculators, screenshots, graphing, spreadsheets, databases, drawing and word-processing software, as appropriate, to enhance mathematical communication.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
There is some appropriate mathematical presentation.
2
The mathematical presentation is mostly appropriate.
3
The mathematical presentation is appropriate throughout.
Criterion C: Personal engagement
This criterion assesses the extent to which the student engages with the exploration and makes it their own. Personal engagement may be recognized in different attributes and skills. These include thinking independently and/or creatively, addressing personal interest and presenting mathematical ideas in their own way.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
There is evidence of limited or superficial personal engagement.
2
There is evidence of some personal engagement.
3
There is evidence of significant personal engagement.
4
There is abundant evidence of outstanding personal engagement.
Criterion D: Reflection
This criterion assesses how the student reviews, analyses and evaluates the exploration. Although reflection may be seen in the conclusion to the exploration, it may also be found throughout the exploration.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
There is evidence of limited or superficial reflection.
2
There is evidence of meaningful reflection.
3
There is substantial evidence of critical reflection.
Mathematics HL guide 69

Criteria and Notes
Additional notes have been written to provide further guidance to moderators on applying the criteria. These notes will provide useful advice to teachers, so are included in this document.
Further advice and information on the exploration is available on the Online Curriculum Centre (OCC), will be included in the subject report, and in the updated Teacher Support Material (TSM). This will include exemplar student work both unmarked and marked.
One area which teachers need to be more familiar with is the use of citations. Information and guidance on all aspects of academic honesty is available on the OCC, but it is essential that students acknowledge sources, and cite these at the point where they occur in the work. It is not sufficient to note sources in the bibliography.
Feedback from moderators indicated that many teachers are not providing comments and annotations on the work. Teachers should provide as much information as possible, including reasons why certain levels are awarded, and background information. Marking information should be included on the work itself, as well as on the form 5/EXCS.
Teachers are also responsible for checking that mathematics used is correct, and to indicate this, or note where it is incorrect.

Criterion A: Communication
This criterion assesses the organization and coherence of the exploration. A well-organized exploration contains an introduction, has a rationale (which includes explaining why this topic was chosen), describes the aim of the exploration and has a conclusion. A coherent exploration is logically developed and easy to follow.
Graphs, tables and diagrams should accompany the work in the appropriate place and not be attached as appendices to the document.
Additional notes
A complete exploration will have all steps clearly explained, and will meet its aim.
Key ideas and concepts should be clearly explained. Mathematical definitions and terminology should be considered under criterion B.
The use of technology is not required (although encouraged where appropriate). Therefore the use of analytic approaches rather than technological ones does not necessarily mean lack of conciseness, and should not be penalised. This does not mean that repetitive calculations are condoned.
An exploration which shows some organisation but does not have some coherence can achieve level 1.
The aim, introduction, rationale and conclusion do not have to be formally identified by the student and may be in the main body of the exploration.
Organisation refers to the overall structure or framework, including the introduction, body, conclusion etc.
Coherence refers to how well different parts of the exploration link to each other. It can also refer to the overall flow, including between different parts, or from text to mathematical presentation etc.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
The exploration has some coherence.
2
The exploration has some coherence and shows some organization.
3
The exploration is coherent and well organized.
4
The exploration is coherent, well organized, concise and complete.

Criterion B: Mathematical presentation
This criterion assesses to what extent the student is able to:
• use appropriate mathematical language (notation, symbols, terminology)
• define key terms, where required
• use multiple forms of mathematical representation such as formulae, diagrams, tables, charts, graphs and
models, where appropriate.
•
Students are expected to use mathematical language when communicating mathematical ideas, reasoning and findings.
Students are encouraged to choose and use appropriate ICT tools such as graphic display calculators, screenshots, graphing, spreadsheets, databases, drawing and word processing software, as appropriate, to enhance mathematical communication.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
There is some appropriate mathematical presentation.
2
The mathematical presentation is mostly appropriate.
3
The mathematical presentation is appropriate throughout.
Additional notes
Mathematical presentation is not the same as communication. However, when there appears to be an overlap, care has to be taken not to penalise a student for the same shortcoming in criteria A and B.
There are multiple facets to criterion B, including using correct notation and terminology, and selecting the appropriate mathematical tool(s) and representation(s).
Level 3 can be achieved by using only one form of mathematical representation as long as this is appropriate.
Consistency in presentation is expected, but if there are inconsistencies which do not adversely affect the use of mathematics, they can be condoned.
Calculator and computer notation should not be penalized if it is software generated. It is expected that students use appropriate mathematical notation in their own work.

Criterion C: Personal engagement
This criterion assesses the extent to which the student engages with the exploration and makes it their own. Personal engagement may be recognized in different attributes and skills. These include thinking independently and/or creatively, addressing personal interest and presenting mathematical ideas in their own way.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
There is evidence of limited or superficial personal engagement.
2
There is evidence of some personal engagement.
3
There is evidence of significant personal engagement.
4
There is abundant evidence of outstanding personal engagement.
Additional notes
There must be evidence of personal engagement seen in the exploration. It is not sufficient that a teacher comments that a student was highly engaged.
There are many ways of demonstrating personal engagement, not just those mentioned in the guide and TSM.
A common “investigation/textbook problem” is unlikely to achieve the higher levels on criterion C unless there is clear evidence that the student has considered the problem from their own viewpoint or other contexts. This could be demonstrated by the students considering and applying new mathematics.
“Abundant evidence” refers to what is reasonable for a DP student (rather than an experienced teacher) to demonstrate in an exploration.

Criterion D: Reflection
This criterion assesses how the student reviews, analyses and evaluates the exploration. Although reflection may be seen in the conclusion to the exploration, it may also be found throughout the exploration.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
There is evidence of limited or superficial reflection.
2
There is evidence of meaningful reflection.
3
There is substantial evidence of critical reflection.
Additional notes
Simply describing results represents limited or superficial reflection. Further consideration is required to achieve the higher levels.
Some ways of showing meaningful reflection are: linking to the aims, commenting on what they have learnt, considering some limitations or comparing different mathematical approaches.
Some ways of showing critical reflection are: considering what next, discussing implications of results, discussing strengths and weaknesses of approaches, and considering different perspectives.
Substantial evidence is likely to mean that reflection is present throughout the exploration. Potentially it may be seen only at the end; however this will need to be of a high quality in order to achieve a level 3.

Criterion E :Use of mathematics
This criterion assesses to what extent and how well students use mathematics in the exploration
Students are expected to produce work that is commensurate with the level of the course. The mathematics explored should either be part of the syllabus, or at a similar level or beyond. It should not be completely based on mathematics listed in the prior learning. If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion.
The mathematics can be regarded as correct even if there are occasional minor errors as long as they do not detract from the flow of the mathematics or lead to an unreasonable outcome. Sophistication in mathematics may include understanding and use of challenging mathematical concepts, looking at a problem from different perspectives and seeing underlying structures to link different areas of mathematics. Rigour involves clarity of logic and language when making mathematical arguments and calculations. Precise mathematics is error-free and uses an appropriate level of accuracy at all times.
Achievement level
Descriptor
0
The exploration does not reach the standard described by the descriptors below.
1
Some relevant mathematics is used. Limited understanding is demonstrated.
2
Some relevant mathematics is used. The mathematics explored is partially correct. Some knowledge and understanding are demonstrated.
3
Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Good knowledge and understanding are demonstrated.
4
Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication expected. Good knowledge and understanding are demonstrated.
5
Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct and reflects the sophistication and rigour expected. Thorough knowledge and understanding are demonstrated.
6
Relevant mathematics commensurate with the level of the course is used. The mathematics explored is precise and reflects the sophistication and rigour expected. Thorough knowledge and understanding are demonstrated.
Additional notes
A key word in the descriptors is “demonstrated”. Obtaining a correct answer is not sufficient to demonstrate understanding. A student must demonstrate their understanding (even limited understanding) in order to achieve level 2 or higher in this criterion.
For knowledge and understanding to be thorough it must be demonstrated throughout the work. Regression using technology is commensurate with the level of the course, but understanding must be
demonstrated in order for the candidate to achieve level 1 or higher.
The mathematics used need only be what is required to support the development of the exploration. This could be a few small topics or even a single topic from the syllabus. It will be better to do a few things well, rather than a lot of things not so well. If the mathematics used is relevant to the topic being explored, commensurate with the course, and understood by the student, then it can achieve a high level in this criterion.

The exploration does not necessarily need to contain the mathematical rigour and sophistication that is assessed in the later parts of the examinations, or that seen in the old IA tasks.
It is acceptable for students to solve hard problems using “simple topics” and “harder topics” may be used to solve simpler problems. In either situation sophistication and rigour can be demonstrated.
The use of technology is highly encouraged, however higher levels are awarded for the sophisticated use of technology as opposed to the use of sophisticated technology.
While topics specifically listed in the Prior Learning are not considered commensurate with the course, other topics not listed in the syllabus may be commensurate.