Choosing a topic and an aim

The choice of topic and aim decides more about the final mark than any other single decision. A well-chosen aim makes every criterion easier: the structure follows from the question, the mathematics has a purpose, personal engagement comes naturally and there is something real to reflect on. A poor choice is hard to rescue by good writing.

Choosing well usually takes several weeks and several false starts. Most students begin with a broad interest, test a few possible questions against the criteria, and end with an aim much narrower than their first idea.

Starting points for a topic

From an interest to the mathematics

The most common starting point is something the student already cares about: a sport, a hobby, a subject studied alongside mathematics, a local issue or a future career. The next step is to ask where mathematics appears in it. A cyclist might ask how gradient and speed are related on a local climb; a musician might ask how the frequencies of a scale are related; a student interested in medicine might ask how the concentration of a drug in the blood changes between doses.

From the mathematics to a context

The opposite route also works well. A student who enjoyed a particular part of the course, such as optimisation, probability distributions or sequences, can ask where it leads next. This route often produces strong mathematics, because the student starts from methods they understand. It needs care to add a genuine question, so that the work does not become a set of exercises.

Keeping an idea log

Ideas rarely arrive on demand. Keeping a list over several weeks, in a notebook or on a phone, gives a much better choice than trying to decide in one evening. Useful sources of ideas include:

• questions that came up in lessons and were not followed up;

• puzzles, games and patterns met outside school;

• data sets that look interesting, such as public transport timetables, sports statistics or weather records;

• other subjects: kinematics in physics, reaction rates in chemistry, population models in biology, elasticity of demand in economics;

• objects that can be measured: the shape of a bridge, the outline of a leaf, the path of a thrown ball.

Testing an idea

Before committing to a topic, test it against five questions. A good topic gives a clear "yes" to each.

The example shows a typical pattern. A vague topic becomes an aim once a measurable quantity is chosen, the variables are named and the method is clear.

From a topic to a focused aim

What an aim is

The aim is a single sentence that states the question the exploration answers. It names the quantity or object being studied, the method or model, and the scope (which data, which cases, which range of values). Some students write it as a research question; others write it as "The aim of this exploration is to…". Either form works.

A focused aim does three jobs. It tells the reader what to expect, it tells the writer what to leave out, and it gives the conclusion something to answer.

Narrowing an aim

Most first ideas are too broad. Narrowing happens in steps, each one removing some of the possibilities.

1. Start with the topic: "music".

2. Pick one mathematical feature: "the frequencies of the notes in a scale".

3. Pick one question about it: "how closely the equal-tempered scale approximates simple frequency ratios".

4. Fix the scope and the method: "To compare the frequency ratios of the twelve-tone equal-tempered scale, $2^{k\mathrm{/}12}$, with the just-intonation ratios such as $\frac{3}{2}$ and $\frac{5}{4}$, and to find which number of equal divisions of the octave up to 53 gives the smallest total error for these intervals."

The final aim is long, but every part of it does work: it names the objects, the measure of quality ("total error") and the range of cases. It also suggests the mathematics (exponents and logarithms, sequences, a measure of error and possibly a proof that $2^{k\mathrm{/}n}$ can never exactly equal $\frac{3}{2}$).

Strong and weak aims in each topic area

Explorations can come from any part of the course. The table gives one weak and one stronger aim from each of the five topic areas.

Each stronger aim is narrow, gives a definite thing to calculate or prove, and leaves room to reflect. The weak aims name a topic only, or ask a question that needs almost no mathematics at the level of the course.

Topics to avoid

Too simple

The mathematics in the exploration must be commensurate with the level of the course. Explorations built only on mathematics from before the course, such as percentages, simple probability with one event, the mean of a data set, area and perimeter, or a straight-line graph, cannot reach the higher levels of criterion E. A topic can start simply, but it must lead to mathematics at course level.

Too advanced

The opposite risk is a topic far beyond the course, such as Fourier analysis, the Riemann hypothesis, general relativity or abstract group theory. A student can usually reproduce the formulas, but not explain why they work. The result is a sequence of quoted results with little understanding, which scores poorly on criterion E, leaves little room for personal engagement and makes meaningful reflection almost impossible.

Purely descriptive

Some topics invite description rather than mathematics: the history of zero, the life of Ramanujan, the golden ratio in art, how public-key encryption works in general terms. These may make a good introduction or motivation, but the body of the exploration has to apply mathematics to a question. A description of the RSA algorithm is descriptive; using modular arithmetic to encrypt and break a small example, then investigating how the time to factorise grows with the size of the key, is an exploration.

Overused topics

Some topics appear so often that it is hard to add anything personal: the Monty Hall problem, the golden ratio in faces, Fibonacci numbers in nature, projectile motion of a ball, the catenary, the mathematics of a Rubik's cube. None is forbidden, and any can be done well, but each needs a genuinely new angle (new data, a new variation of the problem, a new comparison) to show personal engagement.

Data collection

Many explorations need data. Data can be primary (measured, counted or surveyed by the student) or secondary (taken from a published source such as a government statistics office, a scientific database or a sports league's records).

Planning the data

Decide before collecting how much data is needed, how it will be measured and how it will be recorded. Fitting a model with three parameters to five points says very little; testing a probability claim with twenty trials gives an answer dominated by chance. Record the accuracy of each measurement (for example, "lengths measured to the nearest millimetre"), because the reflection will need to discuss how measurement error affects the results.

Ethics

Any exploration that involves people, animals or the environment must be carried out ethically. Schools follow the IB's ethical guidelines for internal assessment, and the teacher should approve the plan before data collection starts.

• Consent. People taking part in a survey or experiment agree to it after being told what it involves. For participants under 18, schools usually ask for permission from a parent or guardian as well.

• Anonymity and privacy. Data are recorded and reported without names, and nothing collected allows an individual to be identified. Sensitive personal information (health, weight, grades, income) is avoided unless there is a strong reason and clear consent.

• No harm or risk. Experiments do not put anyone, including the student, at risk, and do not cause distress, embarrassment or harm to animals or the environment.

• Honesty. All collected data are reported, including awkward values; data are never invented, and outliers are removed only with a stated reason.

Agreeing the aim

Before starting the main work, most schools ask for a short proposal: the topic, the aim, the mathematics expected, the data needed and how they will be collected. The teacher can advise whether the topic is suitable and whether the mathematics is at the right level for SL or HL. The aim can still be refined as the work develops, but a change of direction late in the process leaves little time to finish the mathematics and reflect on it.

Self-check

• To do:

  I have a list of possible ideas, each with the mathematics it would use.

• To do:

  I can test an idea for level, understanding, ownership, focus and data.

• To do:

  I can turn a broad topic into a focused aim that names the quantity, the method and the scope.

• To do:

  I can explain why a topic is too simple, too advanced or purely descriptive.

• To do:

  I can plan data collection with enough data, recorded accuracy and reliable sources.

• To do:

  I can identify the ethical issues in a plan and change it to avoid them.