The Exploration

📋What it actually is

A written report, roughly 12 to 20 pages with double line spacing, investigating an area of mathematics that you chose. Diagrams, graphs and tables count towards the length; the bibliography does not. Around 10 to 15 hours of class and personal time is the expectation.

Note that the page guidance is guidance, not a rule with a penalty attached — the criteria reward quality of mathematical writing, not volume. A tight 13-page Exploration that does one thing properly outscores a sprawling 22-page one that does five things loosely, every time.

🎯The five criteria

The final mark is the sum of five independently applied criteria. Notice how the marks are distributed — it explains a great deal about where to spend your effort.

• A — Presentation (4 marks). Coherent, well organised and concise. That means a real introduction stating the aim, a body that develops logically, and a conclusion that answers the aim. Graphs and tables belong where they are discussed, not banished to an appendix; appendices are for bulk data only. Concise is the word that costs marks: repeated near-identical calculations should be summarised in a table, not printed out ten times.
• B — Mathematical communication (4 marks). Correct notation, symbols and terminology throughout, key terms and variables defined, and more than one form of representation where it helps. Calculator and computer notation is only acceptable if it is software-generated output; typing x^2 or sqrt(x) in your own prose is exactly the kind of thing that pins you at the lower levels. Label every axis on every graph.
• C — Personal engagement (3 marks). Evidence that you made the mathematics your own — thinking independently, presenting ideas in your own way, testing predictions, approaching the topic from more than one angle. It is explicitly not a measure of effort, and a teacher saying you worked hard counts for nothing. A textbook-style write-up of well-known mathematics, however competent, will not reach the top level here.
• D — Reflection (3 marks). Reviewing, analysing and evaluating — and this is the criterion most often thrown away. Merely describing your results is “limited reflection”, worth 1. Meaningful reflection links back to the aim, comments on what you learned, or weighs one approach against another. Critical reflection — discussing the implications of a result, the weaknesses of your method, what you would do next — is worth 3, and it needs to appear throughout, not only in a closing paragraph.
• E — Use of mathematics (6 marks). The largest single criterion. The mathematics must be relevant (it advances the aim), commensurate with the level of the course (not built entirely on prior learning), and correct, with understanding demonstrated. Getting the right answer is not enough: you have to make your reasoning visible. At HL the top levels additionally require sophistication or rigour, meaning mathematics used in a genuinely complex way with claims justified rather than asserted.

🧩Choosing a topic

Three tests a good AA topic passes.

1. It asks a question, not a subject. “Fractals” is a subject and produces a report. “How does the perimeter of the Koch snowflake behave as the area converges?” is a question and produces an investigation. Write your aim as a sentence with a question mark in it before you start.
2. The mathematics sits at course level. Not GCSE algebra dressed up, and not something so far beyond the syllabus that you can only quote results without understanding them. Analysis and Approaches rewards calculus, series, proof, trigonometric identities and functions — use the tools you actually own.
3. You can say something about it that is yours. If the whole report could have been assembled from three web pages, criterion C is already capped.

Directions that work well for AA specifically: modelling a curve you can measure yourself and then justifying the model analytically; investigating the convergence of a series or an iterative process; deriving a result the syllabus states without proof; optimisation of something physical where you set up the constraint yourself; exploring a family of functions and generalising a pattern you find in it.

Directions that reliably disappoint: a history of a mathematician; a summary of a topic already on the syllabus; anything where the “investigation” is one calculation repeated with different numbers; and gambling or lottery topics, which are so common that examiners have seen every version.

📝A workable structure

• Introduction. The aim, in one clear sentence. Why you chose it — specifically, personally, in two or three sentences, not a paragraph of enthusiasm. What you plan to do.
• Background. Define your variables and any terms a peer would not know. State the assumptions you are making, because you will want them later for criterion D.
• The mathematics. Developed in stages, each one motivated by the last. Show the working, not just the results. Where you use technology, say what it did and why that was the sensible tool.
• Testing and extension. Try your result against a case you have not used to build it. Change a parameter and see what survives. This is where criteria C and D are earned simultaneously.
• Conclusion. Answer the aim explicitly. Then evaluate: what were the limitations, what would you do differently, what is the obvious next question.
• Bibliography. Not assessed for quality, but its absence can be flagged as an academic honesty issue. Cite everything, including images, formulae and data you did not generate.

⚠️Academic honesty

The Exploration must be your own work, and you will have to authenticate it. Your teacher may read and comment on one draft — they can tell you where something is wrong or unclear, but they may not correct it for you, and the next version you hand in is final. Collaboration is allowed and encouraged when generating ideas, sharing sources and giving each other feedback on the writing; it is not allowed on a single shared piece of work. If you collect data as a group, you must do your own analysis and say clearly in the report that the data was collected jointly.

The same work cannot be submitted for both the Exploration and the Extended Essay.

📅A sensible timeline

• Well before the deadline: submit a topic and a short outline. Expect the first idea to be too broad; most good aims are the third version.
• Then: do the mathematics before you write the report. Trying to write and investigate simultaneously is what produces reports that wander.
• Draft: the one draft your teacher may comment on. Hand in something complete, not a fragment — feedback on half an Exploration is worth half as much.
• Final: read it once with only criteria C and D in mind, and once out loud for notation. Both passes reliably find marks.