Mathematics

1. What makes Mathematics distinct?

A Mathematics EE must:

• investigate a mathematical question, problem, model or structure

• use mathematics as the primary means of reasoning

• include accurate and appropriate mathematical working

• show understanding, justification and proof where relevant

• present ideas with clarity, precision and logical sequencing

• demonstrate genuine mathematical insight, not just application

Your essay should feel mathematically driven rather than belonging to Economics, Physics or Computer Science.

2. What counts as a suitable Mathematics topic?

A strong topic:

• is based on a mathematical idea or inquiry

• is narrow enough for depth

• allows for meaningful exploration, proof or modelling

• matches the student’s level of mathematical maturity

• may connect with real world contexts, but mathematics must dominate

Examples of suitable topic types

• number theory problems or patterns

• geometry or topology explorations

• mathematical modelling (for example population growth, networks, diffusion)

• fractals, chaos theory or dynamical systems

• cryptography or coding theory

• game theory

• statistical modelling

• combinatorics or graph theory

• analysis of algorithms or optimisation problems

Examples of unsuitable topics

• purely descriptive history of mathematics

• applying simple mathematics to large real world themes

• topics from Physics where mathematical depth is minimal

• essays that rely heavily on software outputs without understanding

• explorations beyond the student’s mathematical capability

• broad overviews rather than precise questions

Mathematics EEs require depth of reasoning, not sweeping generalities.

3. What mathematical work should the essay contain?

Your EE should include:

• clear definitions and notation

• logical development of ideas and arguments

• algebraic, geometric or analytic working where appropriate

• diagrams or graphs if they enhance clarity

• derivations, manipulation, simplifications or proofs

• explanation of how results were achieved

• justification of steps and reasoning

Not required:

professionally formatted typesetting

extremely advanced proofs beyond your capability

What matters is clarity, accuracy and insight.

4. Using sources in a Mathematics EE

Sources should support your exploration, not replace it.

Useful sources include:

• mathematical textbooks

• academic papers accessible at your level

• reputable online mathematical resources

• articles from journals aimed at school or undergraduate mathematicians

Avoid:

• copying long proofs from textbooks

• relying on numerical experimentation without explanation

• using Wikipedia or blogs as your main references

• sources you do not understand

Your essay must show your own reasoning, not a collage of other people’s mathematics.

5. Research methods appropriate for Mathematics

Depending on your topic, methods may include:

• analytical derivation

• algebraic manipulation

• constructing and testing conjectures

• proving theorems

• exploring special cases

• modelling with differential equations

• optimisation methods

• numerical or graphical investigation

• simulations (with explanation and interpretation)

Methods must be mathematically sound and clearly explained.

6. Expectations for analysis

Mathematics analysis should:

• explore ideas in detail

• move beyond simple calculation into reasoning

• connect results to the research question

• identify patterns or relationships

• consider generality or limitations

• interpret results in mathematical terms

A strong EE demonstrates depth of thought and progression of ideas.

7. Expectations for evaluation

Evaluation in Mathematics may include:

• limitations of the approach or model

• assumptions made in modelling

• range and validity of solutions

• complexity or efficiency of a method

• areas for further exploration

• unresolved challenges or conjectures

Evaluation must link logically to your findings.

8. Common pitfalls in Mathematics EEs

Avoid:

• choosing a topic far beyond your ability

• relying heavily on software output (for example Wolfram Alpha)

• copying proofs without understanding them

• producing a descriptive essay about mathematics

• using real world contexts with minimal mathematical depth

• superficial analysis or only performing calculations

• unfocused questions with too many variables

These issues frequently result in weak essays.

9. Model research questions

Here are examples of strong Mathematics EE questions:

1. How can graph theory be used to determine the most efficient route for waste collection in Central Hong Kong?

2. To what extent can fractal geometry model the coastline of Norway?

3. How effective is the logistic map in illustrating chaotic behaviour within population dynamics?

4. In what ways can number theory explain the structure and predictability of the Fibonacci sequence modulo n?

5. How can Fourier series be applied to approximate square waves, and what affects the accuracy of the approximation?

Each question is mathematically rich, focused and explores a specific idea.