How I Beat Cambridge

The science shows that the optimal learning involves active recall, spaced repetition and mixed practice. I worked really hard to structure my study around these ideas, and I think spaced repetition systems are underused when studying for Engineering. In truth, I believe that the principles behind my strategy are sound, and that people would benefit from this style of learning.

This post is my honest, detailed reflection on the spaced repetition techniques that helped me succeed at university. This post is not so much about the specifics of what I did, but rather how I studied in the broadest overview.

Anki

The long term backbone of my study was pretty much entirely done through Anki, with flashcard reviews being my first priority during self study time. I did virtually all of my reviews on the Anki Desktop app, with small number of reviews done on the Ankidroid Android app.

Writing Flashcards

Some of the flashcards I write are shockingly basic. I even wrote a custom Anki note type to speed up the process of churning out the simplest flashcards:

3G4 Medical Imaging Notation
gyromagnetic ratio has the symbol ...

$\gamma$

4M24 Computational Statistics
$\ll$ is the symbol for ...

absolutely continuous

1P1 Thermodynamics
Specific gas constant has the symbol ...

$R$

These cards seem trivial, but they're actually quite significant. Reviewing them gives your brain almost a level of fluency, and scaffolding for higher level ideas to sit on top of. These flashcards are lowest hanging fruit, and are maybe 10% of the flashcards that I write.

If the lecture is a little slow, I also write other basic flashcards. These are things that the lecturer clearly signposts, typically formulated like like "Recall X" or "Define Y":

Recall an expression for the Larmor frequency

$$f = \gamma B_0$$

Recall the formula for the relative entropy

$$ D(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \frac{P(x)}{Q(x)} $$

Recall the formula for the power spectrum density, $S_X(f)$

$$S_X(f) = \sum _{k=-\infty}^\infty R_X(k)e^{-j2\pi fk}$$

With online lectures, you have the time to pause and copy entire proofs as flashcards. These are the kinds of flashcards I write when I'm doing an examples paper, or I'm watching a lecture online:

Simplify $\log \frac{P_{Y^n}\left(y^n\right)}{Q_{Y^n}\left(y^n\right)}$ into into expressions of relative-entropy of an empirical distribution

$$ \begin{gathered} \log \frac{P_{Y^n}\left(y^n\right)}{Q_{Y^n}\left(y^n\right)} =\sum_{i=1}^n \log \frac{P_Y\left(y_i\right)}{Q_Y\left(y_i\right)} \\ =\sum_{i=1}^n \sum_{a \in \mathcal{Y}} \mathbb{1}\left\{y_i=a\right\} \log \frac{P_Y(a)}{Q_Y(a)} \\ =n \sum_{a \in \mathcal{Y}} \frac{1}{n} \sum_{i=1}^n \mathbb{1}\left\{y_i=a\right\} \log \frac{P_Y(a)}{Q_Y(a)} \\ =n \sum_{a \in \mathcal{Y}} \hat{P}_{y^n}(a) \log \frac{P_Y(a)}{Q_Y(a)} \\ =n \sum_{a \in \mathcal{Y}} \hat{P}_{y^n}(a) \log \frac{P_Y(a)}{Q_Y(a)} \frac{\hat{P}_{y^n}(a)}{\hat{P}_{y^n}(a)} \\ =n D\left(\hat{P}_{y^n} \| Q_Y\right)-n D\left(\hat{P}_{y^n} \| P_Y\right) . \end{gathered} $$

Generalize the blue term for multivariate $f(\mathbf x)$ approximating around $\mathbf a$ $$f(x) = f(a) +{ \color{blue} {f'(a)}(x-a) } + \ldots$$

$$\nabla f(\mathbf x)^T(\mathbf x - \mathbf a)$$ (transpose is also acceptable)

Fully simplify $\|\mathbf{A}\|_{2}^{2} =\max \frac{\mathbf{x}^{H} \mathbf{A}^{H} \mathbf{A} \mathbf{x}}{\mathbf{x}^{H} \mathbf{x}} \\$

$$ \begin{aligned} \|\mathbf{A}\|_{2}^{2} &=\max \frac{\mathbf{x}^{H} \mathbf{A}^{H} \mathbf{A} \mathbf{x}}{\mathbf{x}^{H} \mathbf{x}} \\ &=\frac{\mathbf{x}^{H} \lambda_{\max }\left(\mathbf{A}^{H} \mathbf{A}\right) \mathbf{x}}{\mathbf{x}^{H} \mathbf{x}} \\ & =\lambda_{\max }\left(\mathbf{A}^{H} \mathbf{A}\right), \end{aligned} $$

Derive an expression for the Fourier transform of $f'(t)$

$$ \begin{aligned} f^{\prime}(t) & =\frac{d}{d t}\left(\frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d \omega\right)\\ &=\frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \frac{d}{d t}\left\{e^{j \omega t}\right\} d \omega \\ & =\frac{1}{2 \pi} \int_{-\infty}^{\infty}\{j \omega F(\omega)\} \mathrm{e}^{j \omega t} d \omega \end{aligned} $$

Explain what properties of the DTFT lead to the power spectrum density symmetry, $S_X(f) = S_X(-f)$ (2)

$ r_{XX}[k] $ is an even function, implying that the DTFT is real
$ r_{XX}[k] $ is real, is real, implying conjugate symmetry $S_X(f) = S_X(-f) ^*$

Essential tools for writing flashcards:

MathPix. This is one of the few subscription services I bought at Uni and allows me to copy latex directly from screenshots of equations! I bought this partially because I hate crappy screenshots, and partially because I could edit equations to fit the flashcard. I'd really recommend this!
Image Occlusion Enhanced. An addon for Anki, allows you to create "fill in the blank" style image flashcards. Very useful for remembering annotations on diagrams, but I think you can also use these for screenshots of equations?
An Image editor. I used GIMP, but any will do. Useful for basic editing of screenshots that you put in flashcards.

I don't think AI is great at writing flashcards (yet!). Don't get me wrong, AI is fantastic at helping you understand a topic, but AI generated flashcards are often poorly formulated and only cover surface level concepts. Instead, I usually give AI tools clearly defined tasks. I might give ChatGPT the front of a flashcard and get it to generate the back, or ask it to transform existing content, eg. "add colors to this equation to make the proof more comprehensible":

Derive the 'Conditional Smoothing Density' $$ p\left(x_t \mid x_{t+1}, y_{1: T}\right)=\frac{p\left(x_t \mid y_{1: t}\right) f\left(x_{t+1} \mid x_t\right)}{p\left(x_{t+1} \mid y_{1: t}\right)}, t\lt T $$

$$ \begin{aligned} \textcolor{blue}{p\left(x_t \mid x_{t+1}, y_{1: T}\right)} &= \frac{\textcolor{red}{p\left(x_t \mid y_{1: t}\right)} \cdot \textcolor{green}{p\left(x_{t+1} \mid x_t, y_{1: t}\right)} \cdot \textcolor{orange}{p\left(y_{t+1: T} \mid x_{t+1}, x_t, y_{1: t}\right)}}{\textcolor{purple}{p\left(y_{t+1: T}, x_{t+1} \mid y_{1: t}\right)}} \\ &= \frac{\textcolor{red}{p\left(x_t \mid y_{1: t}\right)} \cdot \textcolor{green}{f\left(x_{t+1} \mid x_t\right)} \cdot \textcolor{orange}{p\left(y_{t+1: T} \mid x_{t+1}\right)}}{\textcolor{purple}{p\left(y_{t+1: T}, x_{t+1} \mid y_{1: t}\right)}} \quad \text{(Markov assumption)} \\ &= \frac{\textcolor{red}{p\left(x_t \mid y_{1: t}\right)} \cdot \textcolor{green}{f\left(x_{t+1} \mid x_t\right)} \cdot \textcolor{orange}{p\left(y_{t+1: T} \mid x_{t+1}\right)}}{\textcolor{purple}{p\left(y_{t+1: T} \mid x_{t+1}, y_{1: t}\right) p\left(x_{t+1} \mid y_{1: t}\right)}} \\ &= \frac{\textcolor{red}{p\left(x_t \mid y_{1: t}\right)} \cdot \textcolor{green}{f\left(x_{t+1} \mid x_t\right)} \cdot \cancel{\textcolor{orange}{p\left(y_{t+1: T} \mid x_{t+1}\right)}}}{\cancel{\textcolor{orange}{p\left(y_{t+1: T} \mid x_{t+1}\right)}} \cdot \textcolor{purple}{p\left(x_{t+1} \mid y_{1: t}\right)}} \quad \text{(Cancellation)} \\ &= \frac{\textcolor{red}{p\left(x_t \mid y_{1: t}\right)} \cdot \textcolor{green}{f\left(x_{t+1} \mid x_t\right)}}{\textcolor{purple}{p\left(x_{t+1} \mid y_{1: t}\right)}} \end{aligned} $$

Reviewing Flashcards

After you write your flashcards, you also have to consistently review your cards. It is hard. Spaced repetition systems test you specifically on content you are on the verge of forgetting, and retrieval is challenging. Reviewing flashcards took about 30-40 minutes of focused work most days, and it felt like alot of work.

I wrote a script to visualize my flashcard reviews across four years at Cambridge. I've split out Michaelmas/Lent/Easter term times and vacations into five different blocks to show you how the year went:

First and second years are quite similar: Lectures most of the year and huge exams at the very end. Since flashcards pile up, the workload is naturally heavier in Lent term than in Michaelmas.

I think I got into the swing of things properly in third year. I focused on applying to internships, letting myself fall behind in Michalemas and pick up alot of the slack in Lent. Lent term bioengineering modules are particularly grim, I wrote 500 flashcards for 3G4 alone.

Fourth year has alot of coursework, and I reviewed by far the fewest amount of flashcards here.

Exam practice

Everybody has their own spreadsheet to track exam question progress, and I'm quite fond of mine.

It's nothing really that remarkable. I just have a new row for every tripos question I do, where I draw a color for how well I did in each section of the question. The week before the exam, I would revisit questions I'd previously struggled at.

I plotted some graphs, made a pretty dashboard and found it to be a source of motivation for doing work. I looked at the graph of "number of questions done" pretty much daily in exam season: