End Goal

My aim is to write a single Quarto file that contains both the problems and their solutions that can be rendered into two different documents: one where the solutions are absent, and one where shown in a callout block or otherwise visually distinct section.

Question 1. What is $3\times2$?

Solution 1
We can think of it as ‘three lots of two’, so $2 + 2 + 2 = 6$.

The other restriction is to have a structure that isn’t too technical. Most of the teachers and support staff on this course aren’t programmers, so anything that’s getting into custom Lua filters that will be difficult to maintain is a none starter.

Parametrised Documents

One of the first solutions I tried came from this excellent post by Nicola Rennie. In it she combines Quarto parameter functions with conditional content to create the following solution:

---
...
params: 
  hide_answers: true
---

**Question 1**. What is $3\times2$?

`r if (params$hide_answers) "::: {.content-hidden}"`
 _Solution 1_

We can think of it as 'three lots of two', so $2 + 2 + 2 = 6$.
`r if (params$hide_answers) ":::"`

This solution is nice in that it’s all just R + Quarto and doesn’t require any additional files. To add the styling I was after and make the syntax neater, I did try expanding this with a wee external R function solution.R.

solution = function() {
  if (params$show_solutions) {
    "{.callout-note title=Solution}"
  } else {
    "{.content-hidden}"
  }
}

In the markdown, this would be used as below.

---
...
params:
  show_solutions: true
---

```{r}
#| echo: false
source("solution.R")
```

**Question 1**. What is $3\times2$?

::: `r solution()`
We can think of it as 'three lots of two', so $2 + 2 + 2 = 6$.
:::

However, this was all starting to feel a bit hacky. I couldn’t find whether this approach for constructing divs is intended to work in Quarto, or if it’s a bug. From here, I followed Natalie’s signpost to the assign extension by James J Balamuta.

Quarto Extensions

At face value, the assign Quarto extension was doing exactly what I wanted. After installing the project by

quarto add coatless-quarto/assign

the markdown is simple.

---
...
filters:
  - assign
---

**Question 1**. What is $3\times2$?

:::{.sol}
We can think of it as 'three lots of two', so $2 + 2 + 2 = 6$.
:::

This works by leveraging Quarto profiles, which unfortunately turns out to be the extensions major downside. Neither RStudio or VS Code currently expose profiles through their GUIs, so documents now have to be rendered through terminal commands.

Reading how others worked around this led to finding unilur by Aurélien Ginolhac. Rather than using profiles, it creates custom outputs types; these are supported by RStudio and VSCode.

quarto add ginolhac/unilur

---
...
format:
  unilur-html: default
  unilur-html+solution:
    output-file: example-solution.html
---

**Question 1**. What is $3\times2$?

:::{.unilur-solution}
We can think of it as 'three lots of two', so $2 + 2 + 2 = 6$.
:::

Programmatically this is ideal and very easy to use. Frustratingly though, unilur makes several style choices that are difficult to change:

1. Solutions are collapsed by default, and don’t have an icon to indicate they can be expanded.
2. To avoid overlap, the extension changes the colour of the tip callout, but leaves the icon broken.
3. Headers of solutions use a style distinct to all other callouts, creating a mismatched appearance.

Pure Quarto

It’s around this time, digging into the Lua filters used by assign and unilur to fix these issues, I stopped and asked myself:

Am I overcomplicating this? Could I just have done this in pure Quarto all along?

Fairly early on I’d tried and failed combining the classes {.callout-tip .content-hidden} in a single div, but I hadn’t tried much beyond that before finding Natalie’s post. Exploring more, Quarto also doesn’t seem to allow using conditional spans within the attributes of a div. For example

::: {[.callout-tip]{.content-visible when-meta=answers} [.content-hidden]{.content-visible unless-meta=answers}}
We can think of it as 'three lots of two', so $2 + 2 + 2 = 6$.
:::

just displayed the outer div definition as plaintext, rather than using it as render instructions. This makes sense when I think of the analogy to HTML’s <span> and <div>, though it does reaffirm the uncertainty about running R snippets to define span attributes.

Best I can tell, the only way to do this in pure Quarto is through two nested divs.

---
...
answers: false
---

**Question 1**. What is $3\times2$?

::::{.content-hidden unless-meta=answers}
:::{.callout-tip title=Solution}
We can think of it as 'three lots of two', so $2 + 2 + 2 = 6$.
:::
::::

This is clearly more long winded than ideal, but it’s not doing anything overly complicated and doesn’t require the use of any additional commands or config files. That’s something at least.

Conclusion

I’m not really sure what approach is best, all have their own strengths and weaknesses. Is more verbose Quarto clearer than an R constructor? Is the ease of using an extension worth the compromise on functionality? Hopefully with use in practice (and writing more of these docs with others) a preferred approach will emerge.

What followed was a dive down the rabbit hole of history, travelling from work of the mathematical greats to the depths of Google Group archives, all to answer one question: what’s the deal with the circumference of an ellipse?¹

What’s the problem?

\(\text{Circle Area} = \pi r^2 \\ \text{Circumference} = 2\pi r\)	\(\text{Ellipse Area} = \pi ab \\ \text{Circumference} = \text{???}\)

We’re all taught in school that a circle with diameter \(2r\) has area \(\pi r^2\) and that its circumference is \(2\pi r\). You might have been taught that an ellipse with width \(2a\) and height \(2b\) has area \(\pi ab\), but what about its circumference? It turns out there’s no such ‘elementary’ equation for the circumference of an ellipse.

Why doesn’t it exist?

We remember from school that the arc length of a curve between \(x = x_1\) and \(x = x_2\), given in cartesian coordinates, is

If we first look at circle, when centred on zero² it has equation \(x^2 + y^2 = r^2\), so

We can use the arc length equation to find the circumference, \(s\), of the circle. Using \(x_1 = 0\) and \(x_2 = r\) gives a quarter of the circumference, so

That’s the circumference formula we all know and love! Ellipses are just squashed circles so logically the same should work for them right?

A standard ellipse with width \(2a\) and height \(2b\) has equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Another fact which will be useful later is that the eccentricity of an ellipse is given by \(k = \sqrt{1 - b^2/a^2}\). Working as before,

Oh dear. For ellipses \(\frac{\mathrm{d}y}{\mathrm{d}x}\) is already more complicated than it was for circles, and that’s before we’ve even started integrating. Maybe there’ll be some cancellations though?

Using the equation for arc length as before with \(x_1 = 0\) and \(x_2 = a\) to find a quarter of the circumference gives

at which point we stop and reconsider our life choices. This looks simple but just isn’t going anywhere.

This turns out to be what’s known as the complete elliptic integral of the second kind, a famously non-elementary integral. That is to say, it has no ‘nice’ solutions which means there isn’t an analogue to a circle’s \(2\pi r\) when looking at ellipses. To quote Gérard Michon:

There are simple formulas but they are not exact, and there are exact formulas but they are not simple.³

Showing why it’s non-elementary though goes beyond the scope of this post.

How can it be approximated?

Lets first look at the ‘simple’ formulas that aren’t exact. Over the years there have been many different attempts dreamt up for this, but if we’re wanting truly elementary formulas then we’ve a couple prominent options.

When doing his work on orbits Johannes Kepler (1609) used the geometric average, \(s \approx 2\pi\sqrt{ab},\) to approximate the circumference. It’s very poor with a worst case error of 100%, but it was over 400 years ago so what you gonna do.

About 170 years later, Leonhard Euler (1773) gave it a bash and figured out \(s\) can be bounded below by the arithmetic mean and above by the root mean squared,

More recently, Matt Parker (2020)⁴ debuted his lazy approximation,

which came from computational optimization: finding the numeral rational coefficients for an equation of that form which give the least error on average over a given range of \(a\).

It’s useful to see just how far off the truth these formulas are so we plot their deviation from the true circumference. Without loss of generality, we can vary the value of \(a\) while keeping \(b = 1\); we don’t care about orientation or scale.

Straight away we see that, with the possible exception of Lazy Parker, all these are pretty bad for anything except a very-round ellipse. Euler’s bounds are almost symmetrically bad and Kepler’s attempt shoots way off to the bottom, underestimating by almost 10% already by \(a = 2\).

The Lazy Parker approximation continues to beat these even at the extreme value of \(a = 20\), though by then it is underestimating by 3.43%. If we’re willing to add in some more bells and whistles though we can do better. These aren’t formulas you’d want to memorise but they will get you a good answer.

It wouldn’t be maths if Srinivasa Ramanujan (1914) didn’t have a finger in the pie, and he gave two pretty nice approximations:

1. \(s \approx \pi\left( 3(a+b) - \sqrt{(3a+b)(a+3b)} \right)\),
2. \(s \approx \pi(a+b)\left( 1 + \frac{3h}{10+\sqrt{4-3h}} \right) \text{ where } h = \frac{(a-b)^2}{(a+b)^2}\).⁵

The latter of these is astoundingly good as we’ll see in a moment. Parker also gave an coefficient-optimized solution mimicking the former,

which has decent performance, though at the expense of complexity.

Far more recently, and coming from the unlikeliest of places, David Cantrell (2001) posted the following optimizable formula to a Google Group:

Here \(H_p\) is the Hölder mean. The value of \(p\) can be found through optimization or analysis, with \(p = (3\pi - 8)/(8-2\pi)\) working best for ‘round’ ellipses and \(p = \ln(2)/\ln(2 /(4-\pi))\) being optimal for ‘elongated’ ellipses. In practice, we can use \(p = 0.825\) which does just fine for everything.

Though complicated approximations make for complicated graphs, all six of these approximations outperform all of our simple, rough approximations.

Ramanujan II is an incredible approximation and remains best right through to \(a = 20\), though Cantrell with \(p\) for elongated ellipses beats it soon after.

What if we look even further afield though, maybe at \(a = 75\)? At this point we might be tempted to stop and say

Who’s having an ellipse which is 75 times as wide as it is high? That’s just ridiculous! (Parker, 2020)

After all, even the orbit of Halley’s comet, famed for being highly elliptical, only has \(a = 3.93\) (an eccentricity of \(0.9671\)).

Well it turns out the most elliptical orbit we’ve discovered is of comet C/1965 S1-A Ikeya-Seki⁶. It has an eccentricity of \(0.999915\) giving \(a \approx 76.7 > 75\).⁷

That’s a motivator to look further, up to \(a = 80\). There we see Ramanujan II has started to diverge more significantly from the truth and Cantrell’s approximation is winning out, particularly when

since it’s optimum for elongated ellipses.

All these approximations are pretty good and even the worst among them, Parker’s, stays within 0.5% error. If we want to find the actual circumference of an ellipse though we’ll have to look beyond these approximations. For that we now turn out attention to the exact formulas that are not simple.

How do we find it exactly?

There are three main power series used for ellipse circumferences: one from Colin Maclaurin (1742),

one from Leonhard Euler (1776),

and one from James Ivory (1796)⁸,

While exact on paper these don’t converge in general. We can use them to approximate \(s\) by summing the first \(n\) terms of the power series. For example,

The amended plot below shows that summing Maclaurin’s series to \(n = 6\) is comparable to our rough approximations.

Similarly the plot below demonstrates that summing to \(n = 50\) is comparable to our higher accuracy approximations.

You might then think, okay, to find the exact circumference

Just use a power series to a high number of terms (Disney-Hogg, 2021)

but the practicalities turn out not to be so simple.

The convergence rate of Maclaurin’s series is slow. So slow in fact that the sum to \(n = 10\) terms already starts to diverge from the truth well before \(a = 20\) and is overestimating by 2.38% by the time \(a = 80\). Ivory’s series does fair far better, but for by \(a = 80\) for \(n = 10\) it’s also underestimating by 0.012%.

You might think, maybe we just need higher \(n\)? Well we can push both power series to \(n = 258\) terms⁹ and there is improvement. For Maclaurin it’s hardly the dizzying accuracy we might have hoped for, still being off by about 0.01% at \(a = 80\). That’s barely better than summing 10 terms Ivory, showing just how slow Maclaurin converges.

Meanwhile, even at \(a = 80\), \(n = 258\), Ivory is only producing an error of order \(10^{-11}\%\) which, let’s be fair, is pretty damn good. And that’s not even as far as we can take that series either! If we push it all the way to \(n = 507\) terms⁹, at \(a = 80\) we’re still within machine-epsilon error which is amazing.

In fact at \(n = 507\) we don’t start seeing numerical error until \(a \approx 140\), well beyond our known most elliptic orbits.

Carl Gauss (1812) & Eduard Kummer (1836)¹⁰ showed these three series could all be expressed in terms of the hypergeometric function:

This coupled with the close proximity between the two findings means Ivory’s series is usually referred to as the Gauss-Kummer series.

R’s hypergeo package can evaluate this function, only returning an answer when the terms have converged within machine-epsilon error. Comparisons throughout this work use that function with blind faith that it’s more efficient and accurate than the best power series approximation we were able to use.

How could you do it with Monte Carlo?

The idea behind Monte Carlo integration is simple. To approximate

we first plot \(f\) over \([0, \phi]\). Then we take \(N\) uniformly random points in a box \([0, \phi] \times [0,H]\) enclosing the plot, colouring them red if they fall below the integrand and blue if they fall above it. The integral approximation is then

As the total number of random points increases the approximation converges toward the true value.

At first glance our initial integral

isn’t lending itself to Monte Carlo integration. Not only do we have unknown \(a\) both in the integrand and in the limit, there’s also a lovely asymptote at \(x = a\). It’s not ideal for making a general tool.

On the other hand, the transformation of that integral

is far more inviting. The integrand plot can be enclosed by \([0,~\pi/2] \times [0,~1]\), making it a perfect candidate for doing Monte Carlo integration!

We coded up a simple function in R for doing this for arbitrary \(a\) and \(b\).

MonteCarlo = function(a, b, n=1e3, graph=FALSE) {
    # Given ellipse parameters 'a' and 'b' (where a >= b) and
    # sample count 'n' (default 1000) as inputs, this returns
    # an approximation of that ellipse's circumference using
    # Monte Carlo with n samples. If (optional) bool 'graph'
    # is TRUE (default FALSE) then a graph of the integrand
    # will be plotted.

    # calculate ellipse eccentricity
    k = sqrt(1-(b/a)^2);

    # upper integral limit, calculate 1/4 of ellipse then x4
    phi = pi/2;
    # integrand function, max of this is always 1
    f = function(x) { result = sqrt(1 - k^2 * sin(x)^2); }

    # generate random points in the box
    rand_x = runif(n, 0, phi);
    rand_y = runif(n, 0, 1);

    # true/false vector if each point was under graphs 
    under_integrand = rand_y < f(rand_x);
    # count the number of such points
    number_under_integrand = sum(under_integrand);

    # monte carlo estimation of the area under the graph
    area = phi * 1 * (number_under_integrand/n)

    # optional, graphs integrand and the approximation samples
    if (graph) {
        png("Figs/mc-graph.png")
        par(mar=c(5,5,2,2)+0.1) # remove title space
        # trace out the integrand
        x = seq(0, phi, 0.01);
        y = f(x);
        plot(x, y, type='l', xlim=c(0, phi), ylim=c(0,max(y)))
        # obligatory red/blue MC graph points
        points(rand_x[under_integrand],
               rand_y[under_integrand],
               col="red")
        points(rand_x[!(under_integrand)],
               rand_y[!(under_integrand)],
               col="blue")
        dev.off()
    }

    # final result is 4a times the integral
    return(4*a*area)
}

We’re not the first to apply Monte Carlo to this problem; Gary Gipson (1982) explored the idea more generally as part of his PhD thesis.¹¹

To give a specific example, if we have \(a = 5\), \(b = 1\), our true circumference is \(21.01004\). Carrying out Monte Carlo integration might look as below.

Example graph output from the MonteCarlo function.	Example convergence plot as more samples are added.

In particular, the graph on the left approximates the ellipse’s circumference as

meaning there’s about 2% error. To see how this compares in absolute value terms, the table below has all our approximations all side by side for varying \(a\).

We can see that Monte Carlo isn’t terrible, but it’s also not as good as some of even our more basic approximations like Parker. That begs the question…

Why use Monte Carlo to approximate circumference?

Given Monte Carlo isn’t more efficient, more accurate, or simpler to state than any of the other approximations given here there’s not a clear reason to use it beyond academic curiosity. A better question though might be…

Why use ellipses to demonstrate Monte Carlo?

Monte Carlo approximations aren’t difficult to understand; I’ve a memory of being taught the basics around Y8 (12 years old). We were given a coin to toss 100 times, recording the results and working out the approximate probability of heads by dividing the number of times it happened by 100. It’s not a very exciting demonstration, but it works.

That’s for showing it to a younger audience though, what about when you’re showing it to undergrads? The go-to examples are integration or calculating \(\pi\).

As we did for our elliptic integral, given \(p(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0\) we can approximate \(\int_{x_1}^{x_2} p(x){\mathrm{d}x}\) by generating \(N\) uniform random points in a box bounding \(p\) between \(x_1\) and \(x_2\), counting the number of that fall below \(p\) and dividing that by \(N\). While the idea is still simple, it’s not exactly interesting and feels pretty redundant; polynomial integration can be done exactly and is borderline trivial for most undergrads. Even if \(p\) were to be more complicated, students will be painfully aware there are better approximation methods.

Approximating \(\pi\) has other problems. It’s still integral approximation at heart so it’s not complicated, but it has the aura of a pointless exercise. It does have the advantage of being a well known constant and seeing the approximation converge is a neat demonstration, but how Monte Carlo is being used here is very specific to \(\pi\) and not widely applicable.

Putting aside that it’s not necessarily the best tool for the job, there are some benefits to using ellipse circumference in a Monte Carlo demonstration:

• Unlike approximating \(\pi\), the Monte Carlo integration method used is more widely applicable outside the given context;
• At the same time, unlike \(p(x)\), we don’t have an elementary integral so using an approximation method to find an answer is practical;
• That we don’t have an elementary equation for an ellipse’s circumference is interesting and not a fact all undergrads will have come across;
• The math needed to understand what’s going on is perfectly suited for early undergrads (arc length, integration by substitution, etc), while not being out of reach for enthusiastic sixth-formers.

A downside to bare in mind that it prompts the question, ‘why is this integral non-elementary?’ to which there’s no easy answer; get your references to take higher year integrability courses at the ready!

That aside though, Monte Carlo Elliptic Integrals is a neat tool being applied to an cool problem in a way that’s both interesting and understandable. It’s a good motivator for why Monte Carlo methods might be useful in a practical setting while not presenting an application beyond the understanding of a novice mathematician.

Footnotes & References

Despite the development and maintenance overheads Wren served me well for a good while. Even though the aesthetic was a super basic and the syntax for writing posts was heckin’ clunky, I’d made a surprisingly featureful tool. All the now-standard bells and whistles were there, some of which were state-of-the-art in 2014 like reading times¹ and per tag RSS².

Eventually though I had to let it go. Every time I went to write a blog I had to fix bugs first and it eventually reached the point where I was spending more time on maintenance than writing. In searching for an alternative I found the then-startup Ghost; it used markdown, was open source, and could be self-hosted, promising simplicity so its users could just focus on writing.

I switched to Ghost 0.x and all those maintenance worries melted away.

End of Life

Fast forward to 2021 when Ghost 2.x just reached EOL and 4.x isn’t looking so simple anymore. With the independent explosions of Medium and static site generators, the developers have understandably shifted their focus to bigger multi-author blogs-as-a-business. It’s still self hosted and open source, but that’s a lot of features I won’t be using weighing it down. So what do?

In walks Jekyll. I’d been doing some design work for JustDeleteMe.xyz and had gotten to grips with Jekyll so I could check things were still working on my local build. I’d toyed with it in the past but never went past using one of the default themes. This work with JDM gave me a taste of how easy it was to build on and customise.

Over the last couple of weeks I’ve been tinkering with Jekyll’s Minima theme and my old Wren 0.2 code and they’ve given birth the the wonderful Wren 0.3. It’s a Jekyll theme styled like my old Wren CSS with Liquid files replicating many of the features Wren had.

Shoulders of Giants

While some of these features are novel implementations, most are based on FOSS plugins by other developers. As well as Minima there’s code based on reading time, video embeds, tag cloud, and more. Even the icons are a mix of those I’ve made and modified ones from Font Awesome.

It speaks to a bigger shift in my mentality between 2014 and today. Then I saw value only in building from the ground up; it wasn’t enough to write the blog content, or even the CSS to theme it, I had to write everything.

In 2021 though? I still do think that starting from scratch can be a useful learning exercise and it’s great at teasing out new approaches to problems, but I’m also much more accepting of the benefit of building on the work of other people. There’s no sense reinventing the wheel for the sake of it.

Conclusion

You can find Wren on GitHub to see behind the scenes and find out how it all works. Apologies if the switch to Wren has broke your RSS subscription or old hyperlinks (I couldn’t find a way to redirect those gracefully) but I hope you’ll agree that the refreshed design is worth it!

I’m Josh Fogg (they), a 26 year old math student born in Sheffield in the UK. I did my undergrad masters at Manchester and am now studying for a PhD in Edinburgh. Now we’ve got the GCSE French out of the way, here’s some fun stuff!

Up in Edinburgh I run with the Haries, play dodgeball with the Fireworks, and quidditch with the Kelpies. Quidditch in particular is an amazing and inclusive sport and I do my bit to keep it existing by overseeing volunteer recruitment and HR for QuidditchUK.

In my spare time there are a few open source projects I contribute to, but the big one is Numix where I maintain several icon themes. As well as design, I’m also a jack of all trades when it comes to coding having oodles of experience with Python, C++, MATLAB/Octave, R, and bash, plus some recreational tipples.

Though my dedication to it has fluctuated over the years, I get immense joy out of creative writing. There are a few writing groups I’ve been part of over the years but most of my writing these days is solo. It’s been a while since I was confident in something enough to share it online, here’s to changing that!

All in all, my time ends up divided between quite a range of activities from coding and design to music and gaming. This blog is an amalgamation off all that and an outlet to waffle on and share something that (I hope) you’ll enjoy. So welcome, old friends and new: I hope you enjoy your stay in my little slice of the internet pie! 😃

The pandemic was the defining feature of the 2020/21 academic year, impacting teaching activities throughout. My semester 1 tutoring was delivered through two different ‘hybrid’ models; both involved teaching everyone online whenever COVID restrictions required, but differed in how in-person tutoring was delivered outside that. The first saw tutors work with students in a classroom and online simultaneously, while the second moved students between separate in-person and online groups week-to-week based on need.

Having done online tutoring in 2019/20 semester 2 and over summer with Sutton Trust definitely proved beneficial for the online aspects; expectations for webcam uptake were managed and the necessary tools were familiar, which made focusing on engaging students in the topic easier. Discussions were generally quieter but students did collaborate some on shared virtual whiteboards, an idea taken from a Teaching Cafe ¹. The in-person aspects weren’t so straight forward, with both hybrid models having pros and cons.

The first method was logistically difficult for tutors but easier to timetable. Without live video capture, finding a method of delivery which was engaging for those in person while still accessible to those online proved tricky. After talking with other tutors and trialing setups involving webcams pointed at physical whiteboards, we settled on using Teams’ virtual whiteboard with students in the classroom bringing their own laptops for access. Anecdotal feedback from students in-person unfortunately suggests this still wasn’t a great experience.

The second method gave a better learning environment for those in-person but made timetabling confusing. There was a less consistent group dynamic since attendance changed weekly, but those who attended all semester benefited. Making use of physical space within restrictions did prove problematic; social distancing necessitated more out-loud discussion than normal, when sharing notes might have been possible. This was incorporated into my teaching by delivering chalk-and-talk style workshops but with regular pauses to ask and discuss what would come next in the working.

Another delivery method I tried (inspired by workshops from my undergrad) had students come to the whiteboard to talk through a question they felt comfortable with. I stepped aside while a student talked the group through synthetic division, which is part of the Higher syllabus in Scotland but not usually part of A Levels. On that occasion this worked, but feasibility depended largely on environment (e.g. are disinfectant wipes available, is there enough space, etc).

Semester 2 of 2020/21 was an entirely online affair, but I hope to take learnings from this hybrid teaching, particularly around tools and accessibility, into my 2021/22 tutoring.

Looking at the other national legislatures:

• 0 trans Members of Scottish Parliament (MSPs)
• 0 trans Members of the Senedd (MSs)
• 0 trans Members of the Northern Ireland Assembly (MLAs)
• 1 trans UK Member of European Parliament (MEP)

That’s also not great, but it gets so much worse when you look at the local level. I’ve only been able to find record of there being three trans councilors* ever, one of whom went on to be elected mayor.

This gets so much more egregious when you realise that since there are 20,000 council seats across the UK we’d expect at least 66 trans Councilors at any given time (using Stonewalls lower-bound of there being 200,000 UK trans people). Obviously being trans isn’t something every trans person wants to be open about, but equally to be expecting at least 60 and having only 1 currently is wildly unrepresentative.

This isn’t getting into the politics but I thought I’d highlight it because lack of trans representation at a local level isn’t something I’d thought about before this afternoon.

My research for this clearly didn’t go into enough detail! Thanks to u/Michael273 on Reddit for pointing me to Zoe O’Connell’s brilliant blog which is a much better comprehensive summary of trans representation in UK politics.

In summer 2020 I was given the opportunity to prepare and deliver content as part of a summer school through the Sutton Trust, a charity aiming to improve social mobility in higher education. As a student from a working class background who benefited from Sutton’s work way back in 2012, this was an especially exciting opportunity for me.

Due to the ongoing COVID pandemic many adjustments had to be made to the usual programme. The premise was that for each topic students would be sent a pre-recorded lecture and a problem sheet to complete. They would then get together with the tutors for a video call to talk through the solutions and discuss the topic more generally.

Each topic was developed by pairs of tutors; I covered machine learning alongside Bella Deutsch, with a focus on decision trees. These are taught from SQCF 8 ¹ which fulfilled the criteria of being undergrad-level mathematics students wouldn’t have seen before. To keep it accessible the problems we gave centred around formula usage rather than fully constructing trees from scratch.

With widespread remote teaching still novel there was a lot of uncertainty about how well this new format would work. To learn what was being done elsewhere in the University I attended several staff discussions at the School of Math’s Teaching Cafe ². These proved very useful since small group teaching has been a frequent topic of discussion recently.

As with all online endeavours, the summer school wasn’t without hiccups; on day 1 a system glitch left students without access to the content. This had been resolved by day 3 when we had our call but a significant number of students still hadn’t accessed our topic’s content. Acknowledging that a pandemic perhaps doesn’t make for the best work environment, we adjusted our discussion in the moment to include those who hasn’t done the worksheet. Just because a student didn’t manage to work before the deadline, it doesn’t mean they don’t want to learn; we saw the highest spike in user traffic for the online portion of our worksheet in the day after the video call ended.

Despite last minute changes the discussion group was great fun! Most tutors across the summer school reported difficulties engaging students in verbal discussion (and for safeguarding reasons students weren’t allowed to use webcams), but we had reasonable success getting them to discuss and answer questions in the chat box. After the session was over, two also stuck around to talk (via voice, not chat!) more about AI and the current areas of research.

Teaching online definitely has its hurdles and it’s certainly going to be an exercise in developing new best practices around new modes of delivery. However, as long as we are able and willing to adjust expectations and adapt accordingly we can still provide a quality learning experience to students.

My teaching this year included workshops for first year calculus, working with two groups of about ten University of Edinburgh students. This was intended to be done almost entirely in-person, with students meeting to discuss that week’s problems and submit physical copies of homework. However, events both in the academic community (the UCU strikes from late February to mid March) and the wider world (the COVID-19 global pandemic) meant that it was necessary to do a great deal of adapting to changing circumstances.

In the lead up to the strikes I had honest conversations with students about the issues at hand, which was positively received. I want academia to be a great place of employment for all and the widening casualisation of jobs is an issue I care deeply about. As per UCU’s agreement with the university I continued to mark assessments, and I also took students’ questions by email.

Assessment completion remained pretty consistent during the strikes, prompted by emails from both myself and the course administrator, but engagement with the content which ordinarily would have been covered in workshop dropped significantly. I think this could be due to only passing mentions of the workshops in emails; asking starter questions about that content (e.g. querying understanding of a theorem) may have encouraged more discussion there.

Again when campus closed due to the pandemic I continued to take homework and questions by email, with the addition of a Piazza online forum where students could also discuss the course and ask questions. Learning from the experience of the strikes, this time I managed to engage several students in longer discussions about the course content, helping to develop the ‘how’ and ‘why’ behind methods discussed. This was as simple as giving pointers in my feedback and making it clear more information was available (e.g. “If you would like to know more just let me know!”).

This disruption, particularly the pandemic which saw many students moving across the world, made maintaining tutor-tutee relationships difficult. Several students ran into difficulties around timezones and others from unexpected caring commitments. I endeavoured to make adjustments where possible, providing assistance around work, signposting support channels when appropriate, and checking on students who unexpectedly stopped engaging. The training which we received prior to starting proved helpful here; as well as establishing best practices for delivering workshops, it outlined what is and isn’t within our control as tutors and the importance of that relationship.

Toward the end of the semester one of my students emailed me thanking for the semester and asking for reading recommendations around an area which happened to be my own area of research. We had a chat by email about what I did as a PhD student and I pointed them in the direction of books and course materials which I’ve found useful. It was a heartening end to the term and a welcome reminder of the difference tutoring can make.

2018

Last year I changed tactics slightly and, alongside some traditional big goals, what I really wanted was to form healthy habits. These were tracked using a great little (open source!) app called Loop.¹

Habits

Full disclaimer: for most all habits I didn’t actually meet the ‘85% of days’ goal set. In hindsight focusing on forming the habits was more important meeting an unambiguous (but arbitrary) win condition. With that in mind, this year was still a success in my eyes for most the habits I wanted to form. As the year went on I only got better at sticking to them and this has continued through January.

A big win for me personally has been regular diary updates. It was only sporadically updated in 2017 but by the end of 2018 it was such a regular part of my life that I was actually surprised when I hadn’t updated it the evening before. Looking back at the last two years of entries has also become an unexpected pleasure each night before bed.

Creative writing has definitely been the most difficult to get into, even though it’s something I’ve done a lot in the past. While I’ve occasionally done some scribblings and I had a half hearted bash at NaNoWrioMo I just haven’t made time for it that it needs.

Bigger Goals

On the bigger goals things are a mixed bag, but on the whole looking more positive. Through the tail end of third year I mostly managed to keep on top of assignments and definitely more so going into fourth year. I finished third year with a respectably high 2:1 too; while semester 1 was disappointing I’m happy with how things are going now.

As I say, writing has been difficult. I crashed out of NaNoWriMo pretty early and this blog has been pretty stagnant throughout the year. Reading went better; I didn’t hit the goal, but 13,700 pages over 44 books is nothing to sniff at. This goal was actually on track for most the year, only trailing off in final couple months.

Worries that running a half-marathon would be a low-ball goal were ill-founded. It was incredibly gruelling but on the hottest day of 2018 I completed the Manchester half marathon in a not-terrible ~2 hours time! The goal of reaching out more² was unfortunately less well done for the most part.

2019

Recall that my 2018 goals were split into three broad areas: academic, creative, and health. For 2019 I’m adding a fourth category which is basically my general theme for the year: quitting.

Quitting

For a long time my friends have known me as someone who “does too much”. At pretty much every point in my life I’ve got involved with a load of different activities, for better and for worse. Going into 2019 I want to be more selective with how I spend my time³. It’s not necessarily even that I don’t enjoy some of what I’ll step back from, I just want more space to enjoy what I keep. This is something which started toward the end of last year but will continue I hope.

In addition to carrying on the 2018 habits, I’m adding to Loop several “bad habits” I want to stop or reduce. These include social media time, ordering take out, eating meat, and drinking alcohol.

Bigger Goals Again

On these I’m obviously looking toward to (finally) ending my undergrad and want to be doing something productive come September. I would love to graduate with more than 70% and it’s something that I’ve really worked toward these last few months. Fingers crossed I get there! I’m also hoping to read War and Peace as part of the subreddit r/ayearofwarandpeace.

Making use of my creative outlets is something I want to work on so there’s a hell of a lot on the creative side. I’m reaffirming the goal to publishing something at least once per month with an addition to work on a new icon every day⁴. The goal to read more than 52 books and 15,000 pages is also back, with a slightly bumped up page count. I’d love to complete NaNoWriMo but without knowing what I’ll be doing come November that’s a less certain one.

Continuing the trend of roughly doubling how far I can run each year, now I’m aiming to complete a marathon! My running petered off a little toward the end of last year so looking forward to getting back into it. Similarly did quidditch; at the 18/19 Regionals I played for Manchester’s 2nd team. I really want to get back and make the 1st team for BQC given it could be the last tournament I play at Manchester.

While the quitting theme is more something I’m expecting to work out as I go along, a very know-it-when-you-see-it theme, one big one I already know I’m planning on doing is stepping back from r/up⁵. There’s also an idea for Lent which I’m still musing over.

In short though things are good! There’s a lot to look forward to personally in 2019 and I’m excited to see how it goes.

-Josh