IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example

The table of contents is clear and useful, and the numbered sectioning gives the exploration a strong navigational structure. This helps the reader anticipate the flow from background to methodology to analysis, which supports coherence across the whole report.

The opening aim is focused and mathematically specific: the student states that the investigation will compare both the process and outcome of backpropagation under different activation functions. That clarity gives the whole exploration a clear direction and makes later comparisons easier to follow.

Consider expanding the hidden-layer backpropagation step here rather than only stating that it will "look the same." A brief derivation or worked expression would strengthen the logical chain and make the transition from output-layer gradients to hidden-layer gradients more convincing.

The algorithmic summary is a useful bridge from worked example to computation. The student could strengthen structure further by adding a short sentence explaining exactly what changed between the hand calculation and the automated update, so the reader sees how the two methods connect.

The comparison of convergence rates becomes less coherent here because the method changes across activations: a polynomial fit is used for sigmoid, a power fit for tanh, and a logistic fit for ReLU. The student should justify why these different models are comparable, or use one consistent approach so the comparison is methodologically unified.

The MAU table is a strong comparative device, but the presentation would be even clearer if the student briefly interpreted the ranking directly in the text. Explaining which layer updates were largest and why would make the comparison feel more purposeful rather than just descriptive.

The bibliography still appears to include repeated entries, such as the duplicate Osajima source. The student should remove repeated references and ensure each source appears once in a consistent format, which will make the final section more polished and easier to verify.

The bibliography is extensive and clearly supports the research, but the student could improve presentation by checking that all sources follow one consistent referencing style. Consistency here improves trustworthiness and makes it easier for the reader to trace the mathematical claims back to their sources.

The matrix notation for the network parameters is well chosen and mathematically appropriate for HL work. Using indexed weight matrices and explicit layer labels makes the structure of the network transparent and supports later substitutions cleanly.

The derivative of the sigmoid function is derived step by step using the chain rule, which shows careful mathematical communication. This is effective because the reader can see how the final expression is built rather than simply being given the result.

The simplification to ${\tanh }^{\prime }(x)=1-{\tanh }^2(x)$ is concise and elegant, and it is used correctly in the surrounding explanation. This is a strong example of mathematical notation supporting both efficiency and clarity.

The loss function is introduced with a standard formula and then immediately simplified to the single-example case. That progression is clear and helps the reader understand why the later numerical work can use a reduced expression.

The notation for the error term changes slightly across the derivation, with the text defining ${\delta }_j^{\sigma (x)}$ as a derivative with respect to $Z_k^{[2]}$. The student should keep the subscript and layer indexing consistent so the symbol always refers to the same quantity throughout the backpropagation section.

The ReLU backpropagation notation is not consistently accurate here: the gradient is written with a factor of $Z_k^{[2]}$, but the derivative of ReLU should be expressed as a piecewise indicator, not as the weighted sum itself. The student should rewrite this section so the derivative term clearly matches the activation’s definition.

The worked substitution for the gradient update is a strong communication feature because it shows the exact numbers being used in the rule rather than leaving the calculation abstract. This makes the mathematical process easier to audit and replicates the logic of gradient descent clearly.

What changes in the interpretation if the initial convergence rate is taken at a point other than $x=1$? Asking this would help test whether the comparison is robust or depends too heavily on one arbitrary evaluation point.

Consider adding a brief explanation of why differentiating the fitted curve is a valid proxy for "initial convergence rate." Without that link, the calculation is mathematically neat but the meaning of the derivative in context is not fully explained.

The logistic regression fit for ReLU is stated, but the reasoning behind choosing this particular model is not explained. The student should justify why this curve is the best representation of the data, especially since a different fitting strategy was used for the other activations.

The investigation shows clear personal decision-making in the way the aim is framed around activation functions and backpropagation, rather than simply reproducing a textbook derivation. That helps the work feel purposeful and student-driven.

The methodology reflects independent planning: the student selects a specific training example, describes a step-by-step backpropagation run, and then extends the analysis to full convergence comparisons. That sequencing shows the student is structuring the investigation rather than following a single pre-made example.

The student could show even stronger initiative by briefly explaining why this particular dataset of four examples was chosen. A short justification about representativeness or constraints would make the personal design choices feel more deliberate.

The choice of small random initial parameters is sensible, but the investigation does not test how sensitive the outcomes are to alternative initialisations. The student should vary the starting weights or run multiple trials to demonstrate that the conclusions are not dependent on one convenient setup.

Introducing Mean Absolute Update as an extra comparison metric is a good sign of initiative. It shows the student is thinking beyond final loss alone and trying to capture how the parameter updates differ across layers and activation functions.

To make the investigation feel more original, the student could compare one extra methodological choice, such as a second learning rate or a different activation/output pairing. That would add a more experimental dimension to the current comparison.

How might the conclusions change if the network were trained on a larger, more realistic admissions dataset? Considering this would push the investigation beyond a single worked example and show deeper personal engagement with the modelling context.

The chain-rule setup for the output-layer gradients is mathematically appropriate and clearly aligned to the aim of the exploration. It shows that the student can translate the backpropagation idea into symbolic partial derivatives correctly.

The hidden-layer backpropagation is not actually derived in the main body, even though the discussion claims it will be similar. The student should include at least one explicit hidden-layer gradient expression to show full understanding of the backpropagation process.

The ReLU gradient expression weakens the mathematical accuracy of the exploration because it replaces the activation’s derivative with the weighted sum itself. The student should correct this, since an incorrect derivative undermines both the update rule and the later comparison of activation behaviour.

The numerical loss calculation is correct and well integrated into the backpropagation example. This is effective use of mathematics because the formula is immediately applied to the specific training instance rather than remaining abstract.

The updated matrices show that the student has carried the gradient-descent process through to a full parameter update. This is strong evidence that the mathematics is not just described but actively used to model learning behaviour.

The student could strengthen the mathematics by explaining why the chosen fitted model is differentiable and suitable for gradient interpretation over the interval used. That would make the convergence-rate calculation feel more mathematically justified.

Using a fitted curve and then differentiating it to compare early convergence gives the exploration a more advanced mathematical layer than a simple loss comparison. It is a productive attempt to quantify training speed, not just final performance.

The use of a logistic curve to model ReLU convergence is mathematically questionable unless it is justified against the raw data. The student should explain why this fit is appropriate, otherwise the resulting derivative may not be a reliable measure of convergence speed.

The MAU table could be made more mathematically informative by adding units or a note clarifying whether these are averages over absolute parameter changes per epoch, per update, or per layer. That would make the comparison easier to interpret precisely.