IB Further Maths HL Discontinued: What to Study Instead

Yes. IB Further Mathematics HL was withdrawn in the 2019 curriculum review and was examined for the last time in the November 2020 session. No school can register a candidate for it today, and there is no successor subject carrying that name. The IB Diploma now offers exactly two mathematics courses, each at SL and HL.

What the old course actually was.

Further Mathematics HL was a 240 hour Group 5 subject assessed by two written papers of 2 hours 30 minutes each, 150 marks each, weighted 50 per cent and 50 per cent. It had no internal assessment at all. Almost every candidate took it alongside Mathematics HL, because its six topics (linear algebra, geometry, statistics and probability, sets and relations and groups, calculus, discrete mathematics) assumed the whole Mathematics HL core as prior knowledge.

Why the IB removed it.

The 2019 review replaced four courses, Mathematical Studies SL, Mathematics SL, Mathematics HL and Further Mathematics HL, with two courses at two levels: Mathematics: Analysis and Approaches and Mathematics: Applications and Interpretation. The design principle changed. Students are now separated by the kind of mathematics they need, analytic or applied, rather than by how much of it they can absorb.

The hinge dates.

First teaching of the new courses was September 2019 and first examinations were May 2021. November 2020 was the last Further Mathematics HL session. If a prospectus, a forum thread or an old university offer page still mentions Further Mathematics HL, it is simply out of date, and no amount of self study produces a certificate in a subject the IB no longer examines.

Study Mathematics: Analysis and Approaches HL. It is the direct route for anyone who wanted Further Mathematics HL for the pure mathematics, because proof, complex numbers, vectors, series and differential equations all sit inside its HL only content. If what attracted you was the discrete and statistical side, Applications and Interpretation HL is the right choice instead.

Analysis and Approaches HL in one paragraph.

It is a 240 hour course built on five topics: Number and algebra, Functions, Geometry and trigonometry, Statistics and probability, and Calculus. Assessment is three examination papers plus the mathematical exploration. The HL only content is where the old Further Mathematics flavour survives: proof by induction, proof by contradiction and disproof by counterexample, complex numbers in modulus argument and Euler form with De Moivre's theorem, vector equations of lines and planes, Maclaurin series, l'Hôpital's rule, and first order differential equations solved by separation of variables, by integrating factor, by homogeneous substitution or numerically with Euler's method.

Applications and Interpretation HL in one paragraph.

Also 240 hours, also five topics, also three papers and an exploration, but the HL only content runs through graph theory (adjacency matrices, minimum spanning trees, Chinese postman and travelling salesman problems), matrices with eigenvalues and eigenvectors, Markov chains, phase portraits for coupled systems, and inferential statistics. A graphic display calculator is allowed on every AI paper, including Paper 1.

How the old topics were redistributed.

The table below is the honest mapping. Two of the six Further Mathematics topics were not carried forward at all, and pretending otherwise is the most common mistake in online advice on this question. A full topic by topic walk through of the analytic course is in my IB Maths AA HL syllabus guide.

Mathematics: Analysis and Approaches HL is assessed by three written papers and one piece of coursework. Paper 1 lasts 2 hours, carries 110 marks and no calculator is allowed. Paper 2 lasts 2 hours, carries 110 marks and requires a graphic display calculator. Paper 3 lasts 1 hour and carries 55 marks. The exploration is worth 20 per cent.

The weightings.

Paper 1 is 30 per cent, Paper 2 is 30 per cent, Paper 3 is 20 per cent and the internal assessment is 20 per cent. Further Mathematics HL split its whole grade across two papers of 150 marks each, with nothing internally assessed, so a single bad morning could not be diluted. In AA HL, four separate components share the load, which changes revision strategy: a candidate who is weak without a calculator has 30 per cent of the grade exposed and needs to fix it directly rather than hoping the other paper compensates.

What changed in the shape of the syllabus.

Further Mathematics HL was six compulsory topics with no options. The old Mathematics HL had a core plus one option topic chosen by the school. AA HL has neither: five topics for everyone, no options, and Paper 3 in place of the old option paper. Every AA HL candidate in the world sits the same content.

Conventions that carry across all three papers.

Answers are given to three significant figures unless the question says otherwise, the formula booklet is provided in every paper, and mark schemes award M marks for method, A marks for accuracy, R marks for reasoning, and AG where the answer is given in the question and the working must genuinely reach it.

Proof by induction in Mathematics AA HL is a ladder, not a calculation. You show that you can stand on the first rung, then you show that standing on any rung lets you reach the next one. Everything the mark scheme rewards is the visible link between those two ideas, not the algebra on its own.

Build the intuition first.

The statement you are proving is a whole family of statements, one for every positive integer $n$. You cannot check them one at a time because there are infinitely many. So you check the first one, and then you prove a conditional: if the statement holds for $n = k$, it must hold for $n = k+1$. The conditional is the engine and the base case is the spark. Neither does anything alone, and the IB mark scheme is built exactly on that separation.

Now the exam mechanics.

Proof by induction is HL only content inside Topic 1, Number and algebra, and it appears on Paper 1 and Paper 2 of AA HL. It is normally a full part question. The marks split predictably: M marks for the manipulation of the $n = k+1$ case, A marks for reaching the correct simplified form, and an R mark for the concluding statement. Candidates who do all the algebra and stop lose that R mark every single time.

A worked example.

Prove that for all positive integers $n$ $$\sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4}$$

• Base case: for $n = 1$ the left side is $1^3 = 1$ and the right side is $\frac{1^2 \times 2^2}{4} = 1$, so the statement is true for $n = 1$.
• Inductive assumption: assume the statement is true for $n = k$, that is $\sum_{r=1}^{k} r^3 = \frac{k^2(k+1)^2}{4}$.
• Inductive step: $\sum_{r=1}^{k+1} r^3 = \frac{k^2(k+1)^2}{4} + (k+1)^3$, and taking out the factor $\frac{(k+1)^2}{4}$ gives $\frac{(k+1)^2}{4}\left(k^2 + 4(k+1)\right) = \frac{(k+1)^2(k+2)^2}{4}$.
• Conclusion: that is the original formula with $n$ replaced by $k+1$, so truth for $n = k$ implies truth for $n = k+1$. Since the statement is true for $n = 1$, by the principle of mathematical induction it is true for all positive integers $n$.

What the examiner is actually reading for.

The one line where the assumption gets used. A proof that jumps straight from $\sum_{r=1}^{k+1} r^3$ to the closed form, without ever writing $\frac{k^2(k+1)^2}{4} + (k+1)^3$, has not used the assumption, and the method marks go with it. Write the assumption down, then quote it in the next line. This habit of making the logical step visible is the same one that decides marks across the whole HL course, and it is one of the reasons AA HL feels harder than students expect, as I explain in the real difficulties of IB Maths.

No. Since the final November 2020 session, university offers for mathematics, engineering and physics are written in terms of Mathematics: Analysis and Approaches HL. Selective courses commonly specify a 6 or a 7 in AA HL, with 7 the usual requirement at the most competitive mathematics departments. No admissions office can demand a subject the IB has stopped examining.

What Further Mathematics HL used to signal.

It told an admissions tutor two things: that the candidate had seen mathematics beyond the standard HL core, and that the school was willing to teach it. Both signals now have to come from somewhere else. The first comes from the AA HL grade itself and from Paper 3 performance, since Paper 3 is the extended problem solving component. The second comes from the exploration, where the choice of topic shows what a student can do when nobody sets the question.

What to put in your subject choices instead.

For a mathematics or physics degree, the standard combination is Analysis and Approaches HL with Physics HL and a third HL that fits your interests. For engineering, Analysis and Approaches HL is expected by most departments, although some accept Applications and Interpretation HL, so check every course page individually rather than assuming. I have written the full comparison in AA HL versus A-Level Further Maths for STEM universities.

One warning about old offer pages.

University websites are slow to update. A page that lists Further Mathematics HL as desirable was written before 2021 and has not been reviewed. Contact admissions rather than trying to satisfy it, because there is literally no way to satisfy it now, and no university treats an impossible requirement as binding.

Use Paper 3 and the exploration. Paper 3 in Mathematics AA HL lasts one hour, carries 55 marks and 20 per cent of the final grade, and contains two compulsory extended response questions that build unfamiliar mathematics in stages. The exploration is worth another 20 per cent and lets you take a topic as far as you can defend it.

Why Paper 3 is the real successor to the option paper.

Each Paper 3 question starts from something concrete, then generalises, then asks you to prove or justify the pattern you have found. Parts are chained: a question that says hence is telling you that the previous result is meant to be used, and a question that says hence or otherwise is telling you it is the fastest but not the only route. A graphic display calculator is allowed. The single most useful practice is doing a whole Paper 3 question in one sitting without looking at the mark scheme between parts, because the difficulty is sustained reasoning, not any individual step.

Why the exploration is where the lost topics can come back.

The mathematical exploration is marked out of 20 across five criteria: presentation (4 marks), mathematical communication (4 marks), personal engagement (3 marks), reflection (3 marks) and use of mathematics (6 marks). Criterion E is the one that rewards ambition, since at HL the mathematics must be commensurate with the level of the course and be correct. Group theory, graph theory, numerical methods, error analysis: all of it is legitimate exploration territory, and none of it appears on an AA HL paper.

The trap.

Sophistication only counts once the mathematics is correct and genuinely relevant to your research question. An exploration that quotes a theorem it never uses scores worse than one that works a modest idea through properly. The criteria and the process are set out in my IB internal assessment guide.