Threshold concepts, learning and Pokémon

Last academic year I took a course on teaching and learning in higher education. I enjoyed learning some education theory: I could recognise habits (both good and bad) my students and I practised. I wanted to write up some of the more interesting ideas I came across, I’ve been kept busy by other things (such as writing up the assessment for the course), but here’s the first.

Pokémon PhD

My collection of qualifications.

Threshold concepts

Have you ever had that moment when something just clicked? Perhaps you’ve been struggling with a particular topic for a while, then suddenly you understand, you have that eureka moment, and you get a new view on everything. That’s one of the best moments in studying.

Threshold concepts are a particular class of these troublesome concepts that have a big impact on your development. It’s not just that these take work to come to grips with, but that you can’t master a subject until you’ve figured them out. As a teacher, they’re something to watch out for, as these are the areas where students’ progress can be held up and they need extra support.

Being a student is much like being a Pokémon. When you start out, there’s not much you can do. Then you practise and gain experience. This can be difficult, but you level up. (Sadly, as a student you don’t the nice little jingle when you do). After levelling up, things don’t seem so hard, so you can tackle more difficult battles. Every so often you’ll learn a new technique, a new move (hopefully you won’t forget an old one), and now you are even more awesome.

That’s all pretty straightforward. If you keep training, you get stronger. (It does turn out that studying helps you learn).

Mastering a threshold concept is more like evolving. You get a sudden boost to your abilities, and now you can learn moves that you couldn’t before, perhaps you’ve changed type too. Evolving isn’t straightforward. Sometimes all you need to do is keep working and level up; other times you’ll need a particular item, to learn a special move, to hone one particular aspect, or be in the right place at the right time. Some people might assimilate a threshold concept like any other new idea, while others will have to put in extra time and effort. In any case, the end effect is transformative. Congratulations, your Physics Student has evolved into a Physicist!

Do di do dum-di-dum-di-dum!

Educational evolution. Pokémon art by Ken Sugimori.


Every discipline has its own threshold concepts. For example, in Pokémon training there’s the idea that different types of Pokémon are have advantages over others (water is super effective against fire, which is super effective against grass, etc.), so you should pick your Pokémon (and their moves) appropriately. Threshold concepts share certain attributes, they are:

  • Transformative: Once understood they change how you view the subject (or life in general). Understanding Pokémon types changes how you view battles, if you’re going to go up against a gym leader called Lt. Surge, you know to pack some ground types as they’re good against electric types. It also now makes sense how Iron Man (obviously a steel type), can take on Thor (an electric type) in The Avengers, but gets trashed by some random henchpeople with heat powers (fire types) in Iron Man 3.
  • Irreversible: Once learnt there’s no changing back. You know you’re going to have a bad time if you’ve only packed fire types to go explore an underwater cave.
  • Integrative: Having conquered a threshold concept, you can spot connections to other ideas and progress to develop new skills. Once you’ve realised that your beloved Blastoise has a weakness to electric types, you might consider teaching it Earthquake as a counter. You’ve moved on from just considering the types of Pokémon, to considering their move-sets too. Or you could make sure your team has ground type, so you can switch out your Blastoise. Now you’re considering the entire composition of your team.
  • Troublesome: Threshold concepts are difficult. They may be conceptionally challenging (how do you remember 18 types vs 18 types?), counter-intuitive (why don’t Ghost moves affect Normal types?), or be resisted as they force you to re-evaluate your (deep held) opinions (maybe Gyarados isn’t the best, despite looking ferocious, because it has a double weakness to electric types, and perhaps using your favourite Snorlax in all situations is a bad idea, regardless of how huggable he is).

Using these criteria, you might be able to think of some threshold concepts in other areas, and possibly see why people have problems with them. For example, it might now make more sense why some people have problems accepting global warming is caused by humans. This is certainly a transformative idea, as it makes you reconsider your actions and those of society, as well as the prospects for future generations, and it is certainly troublesome, as one has to accept that the world can change, that our current lifestyle (and perhaps certain economic activities) is not sustainable, and that we are guilty of damaging our only home. The irreversible nature of threshold concepts might also make people resist coming to terms with them, as they prefer their current state of comfortable innocence.

Loss of Arctic ice over 15 years

National Geographic atlases from 1999 to 2014, showing how Arctic ice has melted. At this rate, ice type Pokémon will be extinct in the wild by the end of the century (they’re already the rarest type). It’s super depressing…


Threshold concepts are key but troublesome concepts within a discipline. If you want to be the very best, you have to master them all. They are so called as they can be thought of as doorways, through which a student must step in order to progress. After moving passed the threshold, they enter a new (larger) room, the next stage in their development. From here, they can continue to the next threshold. Looking back, they also get a new perspective on what they have learnt; they can now see new ways of connecting together old ideas. Students might be hesitant to step through because they are nervous about leaving their current state behind. They might also have problems just because the door is difficult to open. If you are planning teaching, you should consider what threshold concepts you’ll cover, and then how to build your lessons around threshold concepts so no-one gets left behind.

I especially like the idea of threshold concepts, as it shows learning to be made up of a journey through different stages of understanding, rather than building a pile of knowledge. (Education should be more about understanding how to figure out the right answer than knowing what it is). If you’d like to learn more about threshold concepts, I’d recommend browsing the resources compiled by Michael Flanagan of UCL.

Perks and perils of a PhD

Nijō Castle garden.

Pond in the gardens of Nijō Castle, Kyoto. A good spot for pondering. The castle has whistling floorboards, to warn you if an assassin is sneaking up on you. Modern buildings don’t do enough to warn you of assassins.

This blog has been neglected recently as I have been busy travelling, with conferences and meetings (with a little holiday in between) in Japan, Korea and Germany. I am now back in Birmingham where we have a veritable army of summer students. They are all enthusiastic, and seem to be doing well at both their projects and joining in lunchtime conversations. One asked whether it was a good idea to do a PhD? Travelling to interesting places in one of the perks of being an academic, but does it compensate all the hard work? Here are my thoughts on doing a PhD now mine is safely done but still fresh in my memory.

The third degree

A PhD is not a simple continuation of your studies. One of the things that surprised me was how different research is from study (although they may share many of the same skills). At school and undergraduate you learn: you pay attention in class, you do assignments and projects, you revise and you take assessments. If you work hard at these, you pick up new knowledge and skills, and end up doing well. (Wooh!) In research, you have to solve problems, to figure out how to do things that have never been done before (which may require picking up new knowledge and skills). This can be extremely exciting: you could be the only person in the world to know how to do something, but since you are trying something new, it could also turn out not to work… You can work hard in a particular area for days, weeks or even years, and it all come to nothing.

Research projects at an undergraduate level are different from those at postgraduate. The former are usually designed to be safely solvable, and even if things don’t work out, you come to the end of your time and be given marks for effort. It’s much harder to put together a PhD dissertation without results, and a lack of progress (perhaps especially if through no fault of your own) can be especially demotivating.

When asked about doing a PhD, the current PhDs showed varying levels of enthusiasm. This is usually correlated with how things are going and how close they are to finishing. Maggie, who is always keen on encouraging people to learn about science, has put together a list of 5 reasons why you should do a PhD. I think these neatly sum up some of main motivations for doing a PhD.

1. Freedom and flexibility

Academia enjoys a lot of freedom. And I don’t just mean that you don’t have to wear a tie

You don’t have to work standard office hours, but can often schedule things around you. This can be especially handy if you have family commitments, you don’t function well in the morning, or just fancy an afternoon off. It can also have the downside of blurring the work/life divide. Working from home gives you the flexibility to work in your pyjamas, but it can also makes it easy to work evenings and weekends—perhaps the weekends are the best time to come in to the lab because there are fewer people trying to use the most shiny equipment. It can be difficult to maintain a healthy work/life balance, and it can also lead to ridiculous expectations that you can work all the time (or should feel guilty if you’re not). Of course, sometimes you have to visit the lab every two hours to look after your experiment, and there’s no flexibility at all.

As well of freedom in when you schedule work, there is also freedom in what you do. It’s difficult to predict where a PhD will go, but you can focus in on what you are interested in and what you enjoy. Your supervisor, future examiners and potential employers may disagree with you about what’s worthwhile researching, so you do have some constraints; however, as long as solve an interesting problem, it doesn’t matter as much as in industry if it’s a different one to the one you started with. Some of the best PhD projects I have seen (or been involved with) come about because the student came across a new technique they wanted to play with, read up on a different area out or just wanted to help answer someone else’s question. Procrastination can have some useful side-effects.

2. The title

Being a doctor is pretty cool. Not as cool as being the Doctor, but still, it can command some respect. However, that doesn’t mean you receive (or deserve) any special treatment. Contrary to popular opinion, your title doesn’t go on your passport. It does indicate that you are an expert in one particular area; however, this might be an obscure and unhelpful one.  If you are ever on a flight and the attendant asks “Is there a doctor of astrophysics on board?” you are probably sufficiently doomed that you might as well just stay seated and try to finish up your peanuts in whatever time remains.

In the end, it is having completed the difficult research and produced a quality thesis that is worth the respect, and not the extra letters with your name. If you are not interested in the the former, the latter will not give you the motivation to put in the time and effort to complete it.

3. To prove you’re smart

Leading on from the last point, a doctorate is a seal of academic quality. However, I really wouldn’t suggest doing a PhD because you need to validate your intelligence. You are intelligent whether or not you decide to go to graduate school, and one should never assume that someone is less smart because they lack a PhD—first, because you do not know what opportunities they may or may not have had in life, and second, intelligence is about more than academic achievements. If you’re a lazy writer, giving a character a PhD is an easy way to establish they are clever without having to think of way for them to show it. In real life, people will soon figure out how smart you are by interacting with you (if they are only interested in titles, find someone else).

Getting a PhD isn’t just a case of being smart, it’s not a prize on a game show. As much as intelligence, a doctorate requires determination. Undergraduate is like a sprint, you can work really hard for a short stretch (around exams) and then collapse. The quickest people will come out on top, but you could still take it at a jog and make it to the end provided you don’t mind being second. A doctorate is more like a marathon, its not enough to be fast, you need to be able to keep going, to pace yourself and to pick yourself up if you trip. Both can be exhausting and painful, but it’s much more important to figure out if you can really go the distance before starting on the 26 miles.

Perhaps you are unsure if you’re up-to-scratch and want to try a PhD to see? Finding out that you can do it may be a huge confidence boost! However, academia can also batter your ego as you’ll be surrounded by other equally intelligent people. I guess you just need to be happy with who you are.

Finally, parents may like to show off the achievements of their children, and you may make your friends proud, but it’s not them who have to spend the time in the library. Making people you love proud is wonderful, but so is spending time with them. A PhD can consume huge amounts of time, energy and attention (especially while writing up). It should be something that you want to do, not something other people want you to do.

4. Broadening your horizons

Academia does give you the chance to visit new places and work with people all over the world. I really enjoyed my summer travels and Maggie is currently observing at a telescope in Chile. Of course you don’t always have that luxury: sometimes conferences aren’t in interesting places or funding could be running short. My first two conferences were in Glasgow and Cardiff. Both are lovely cities to visit, but neither was a once-in-a-lifetime opportunity. If you are really keen on travel, then there are other careers that give you better excuses to travel. Or you could take a better paid job and just pay for yourself to go on holiday. Travel, like free coffee, is a perk, but it’s not enough to justify doing a PhD.

The KAGRA detector

I visited the in-construction Kamioka Gravitational Wave Detector (KAGRA) in Japan. It is being built underground, in an old mine in the Hida Mountains. You can see part of the vacuum tubing for one of the laser-interferometer arms in the foreground, and where they’re going to suspend a mirror from the room above in the background. It’s amazing engineering and the views outside are impressive too! They’re on a tight schedule, aiming for a first run (albeit at terrible sensitivity) this year.

More importantly, a PhD gives you the opportunity to come across new ideas and ways of looking at a problem; to work with interesting, intelligent people from a range of backgrounds, and time to examine the world (or Universe) in detail in many different ways. That might all be from your cluttered desk, but it can be really exciting.

5. For knowledge

Over the course of a doctorate you will learn many things: the best seminars for free food, how to manage you supervisor and lots of transferable skills. However, the big thing is your thesis. Through your research, you will contribute something to the sum of human knowledge. It may be revolutionary, it’s more likely to be something that will go towards a bigger question (with help from lots of others), but it could also be the discovery that this particular thing doesn’t work. You research will push back the boundary of the unknown. You will become a world expert in your area: no-one will know your research as well as you do. If there is a topic that really interests you, if there is something that you want to know more about, then a PhD gives you the chance to explore this.

In my opinion, this is the only reason to do a PhD. There are other benefits and perks, but this should be your motivation. A PhD is not just a training course, but is another step towards understanding everything. I think that is amazing.

The forbidden motivation

Having been through the list, you may think there is something missing. What about doing a PhD to get a job? There are few careers that require a PhD, and it may not serve any more advantage than a Masters. Doing a PhD probably won’t make you rich. It may make you more attractive to some employers, but maybe spending the same amount of time working your way up from a lower rung would be just as effective? Extremely few employers have any kind of hiring scheme for PhDs, so in many cases you would start at a similar level as someone with an undergraduate degree. Some areas, of course, have strong industrial links, so it’s easy to move across. In this case, doing a PhD can be a great option: you can even get to work with potential future employers during your study (and possibly get some extra funding). The usefulness of a PhD strongly depends on the area.

There is one domain where a PhD is the well-established first step. Academia. Many think of academia as the logical progression, but it is not. You are not guaranteed an academic job with a PhD. In the sciences, most PhDs will not continue in academia. According to a report from the Royal Society, only 3.5% of science PhDs in the UK end up with a permanent academic position, and only 0.45% become professors. Competition is extremely tough: the number of PhDs awarded is increasing rapidly, but the number of faculty positions is remaining constant. I do not think the situation is better in the arts. A PhD student should not expect to get a job in academia.

The answer

A PhD is a big commitment, and requires careful contemplation. There are many reasons why you might be considering doing one; however, I think that if you’re going to enjoy the experience, the motivation you need is the desire to spend several years getting to know one particular area really well. You must be happy to invest years of your life without any guarantee of returns. You’ll pick up many useful skills, but that will not make you irresistibly desirable to employers—some will regard you as overqualified, and the prospects of an academic career are slim. You will receive opportunities you wouldn’t otherwise, in particular, to meet some awesome people. A PhD is a challenge, research can be both deeply rewarding and excruciatingly frustrating (sometimes in the same afternoon). On balance, if it is the right thing for you is deeply personal. As is common in research, to answer this question, we need further data.

Should you decide to go for it, the next thing to think about is the area and the location. You should make sure you get adequate funding, and take care in picking a supervisor—always talk to their current students. Perhaps we’ll come back to these points later. Good luck!

Lanterns at the Jogyesa temple, Seoul.

Panorama of lanterns at the Jogyesa temple, Seoul. Beautiful, and you can keep following it around in circles, just like a PhD…

A level subject choices

Ofsted have recently published statistics relating to the subject choices of students starting A levels in England in 2013/2014. (For those unfamiliar with A levels, they are the qualifications taken between the ages of 16 and 18, students usually pick 3–4 subjects for the first year, which is known as AS, and normally slim down to 3 for the second year, A2; university admissions are based upon A level results). This is part of an effort to understand what drives students to pick different subjects and particularly science. Engaging students in science is a challenge, although many enjoy it or can achieve well in tests, then can struggle to see that it is for them. In Physics, we have a particular problem recruiting girls, which means we are not getting the best mix of people. I was interesting in having a look at the subject choices, so I’ve put together a few graphs.

Subject popularity

The most popular subjects at AS level are:

  1. English,
  2. Mathematics,
  3. Psychology.

English and Maths make sense, as they’ll be familiar from previous study and are of general applicability. I was surprised that Psychology came third, since it’ll be a new subject; the top ten consists of subjects familiar from pre-16 education, with the exception of the two social sciences, Psychology and Sociology (8). Physics comes in at number 7, behind both Biology (4) and Chemistry (6). This makes me sad, but at least Physics is still one of the most popular choices.

The distribution of student numbers is show in the graph below. I’ve not quite figured out what the distribution of student numbers should be, but it’s roughly exponential. There are too many subjects to label individually, so I’ve grouped them roughly by subject area. The main sciences (Biology, Chemistry and Physics) all do rather well, but modern languages are languishing towards the bottom of the list (top is French at 21). The smallest subjects have been grouped together into Other categories, these make up the bottom of the distribution, but in amongst them are Classical studies (29), German (30), and Accounting & finance (31).

Subjects ranking

Student numbers in the most popular subjects at AS level (in England 2013/2014). Data from A level subject take-up.

Gender differences

The report also lists the numbers of boys and girls taking each subject. I know that Physics is male-dominated, but I didn’t know how this compared to other subjects. To quantify the imbalance, I’m going to define the asymmetry as

\displaystyle \mathrm{Asymmetry} = \frac{\mathrm{No.\ of\ girls}\ -\ \mathrm{No.\ of\ boys}}{\mathrm{No.\ of\ students}}.

This is 0 if there are equal numbers of boys and girls, and is ±1 if completely made up of boys (−1) or girls (+1). Overall, more girls than boys are taking A levels, giving an total asymmetry of 0.0977. That’s not great, but we’ll see it’s smaller than is typically the case for individual subjects.

The most male-dominated subjects are:

  1. Computing (−0.8275),
  2. Physics (−0.5446),
  3. Further mathematics (−0.4569).

The most female-dominated subjects are:

  1. Sociology (0.5084),
  2. Art & design (0.4896),
  3. French (0.4531).

We see that Physics is in pretty poor shape, being the second most asymmetric subject overall. However, Computing is really out in a league of it’s own: there are almost 11 boys for every girl in the subject! That is not healthy. The most balanced subjects are:

  1. Geography (0.0056),
  2. Chemistry (−0.0167),
  3. Government & politics (−0.0761).

These are the only subjects with asymmetries smaller than the overall population of students. The gender balance in Chemistry shows that the Physical sciences don’t need to be male-dominated; however, this could equally reflect the compromise between male-dominated Physics and female-dominated Biology (0.2049).

The graph below plots the number of students taking a subject and its asymmetry. There’s no real trend with student numbers, it’s not the case that it’s easier for smaller subjects to become biased or that it’s easier for larger subjects to develop a reputation.

Asymmetry and number of students

Scatter plot of the number of students and gender asymmetry of AS subjects (in England 2013/2014). Higher points are more female dominated and lower points are more male dominated. The dashed line indicates gender parity and the dotted line indicates the average for all subjects. Data from A level subject take-up.

Normally, I’d expect there to be scatter in a quantity like asymmetry: some values high, some low, but more clustering in the middle than out in the extremes. Looking at the plot above, this doesn’t seem to be the case. There are relatively few subjects in the middle, but there seem to be two clusters, one at small positive asymmetries and another at small negative asymmetries. I’ve plotted the distribution of subject asymmetries below. To make it clearer to view (and to make a nice smooth, continuous distribution), I’ve smeared out the individual subjects. These means I’m actually plotting the density of subjects per unit of asymmetry, rather than the number of subjects: if you work out the area under the curve, that gives the number of subjects in that range. (For those who care, I’ve convolved with a Gaussian kernel with a standard deviation of 0.1, and made sure to renormalise them so that the total area is correct).

Asymmetry distribution.

Smoothed distribution of gender asymmetry for AS subjects (in England 2013/2014). Left is male dominated and right is female dominated. The area under the curve gives the number of subjects. The diamonds mark the locations of individual subjects. Data from A level subject take-up.

It does appear that there are two peaks: one for boys’ subjects and another for girls’. Computing is off being a clear outlier. However, if I turn up the smoothing (using a standard deviation of 0.3), this disappears. This always happens if you smooth too much…

Asymmetry distribution.

Heavily smoothed distribution of gender asymmetry for AS subjects (in England 2013/2014). Left is male dominated and right is female dominated. The area under the curve gives the number of subjects. The diamonds mark the locations of individual subjects. Data from A level subject take-up.

It looks like this is one of the cases where I should really do things properly and I should come back to look at this again later.

Regardless of whether my suspicion of there being two clusters of subjects is correct, there does appear to be a spectrum of subjects, with some being as perceived as for boys and others for girls. This differentiation exists already exists at age 16—even for subjects like Psychology and Sociology that have not been studied previously. It seems that these stereotypes are ingrained from an earlier age.

Computing and Psychology role models

Ada, Countess of Lovelace, mathematician and first computer programmer (and superheroine), and Sigmund Freud, neurologist and founder of psychoanalysis. Evidence that there really shouldn’t be divides in Computing, Psychology or any other subject.


As well as looking at how many students take AS, we can look at how many continue to A2. The report gives the percentage that continue for both boys and girls. The distribution of all continuation percentages is shown below, again with subjects grouped by area. The average progression across all subjects is 72.7%.

Continuation ranking

Percentage continuation from AS to A2 for different subjects (in England 2013/2014). The dotted line indicates the average. Data from A level subject take-up.

The top subjects for continuation to A2 are:

  1. Other modern languages (90.4%),
  2. Drama (82.7%),
  3. Media/film/TV studies (81.4%).

Other modern languages is the smallest subject in terms of student numbers, but has the highest continuation: I guess those who opt for it are dedicated to seeing it through. However, there doesn’t seem to be a correlation between student numbers and continuation. English, the most popular subject, comes in just below Media/film/TV studies with 81.2%. The bottom subjects for continuation are:

  1. Other social sciences (45.9%),
  2. Accounting & finance (59.7%),
  3. Computing (61.4%).

I don’t know enough about these subjects to know if there might be a particular reason why just taking them for one year might be useful. In contrast to Other modern languages, German (62.7%), French (64.1%) and Spanish (65.8%) have some of the lowest continuation rates (coming in just above Computing). Physics also does poorly, with only 67.8% continuing, below both Chemistry (71.0%) and Biology (72.2%). For comparison, Further mathematics has 68.3% continuation and Mathematics has 75.4%. I would expect continuation to be lower for subjects that students find more difficult (possibly with the biggest jump from GCSE).

Now, let’s have a look at the difference in progression between the genders. In the figure below, I plot the difference in the percentage progression between boys and girls,

\mathrm{Difference} = \mathrm{Percent\ girls\ continuing}\ -\ \mathrm{Percent\ boys\ continuing},

versus the asymmetry. The two quantities show a clear correlation: more girls than boys progress in subjects that are female dominated and vice versa. Gender asymmetry gets worse with progression.

Asymmetry and progression

Scatter plot of the gender asymmetry and difference in percentage progression of AS subjects (in England 2013/2014). Left is male dominated and right is female dominated. Higher points have a higher proportion of girls than boys continuing and lower points have a higher proportion of boys than girls continuing. Data from A level subject take-up.

The subjects with the largest differences in continuation are:

  1. Physics (−14%),
  2. Other science (−12%),
  3. Psychology (11%).

That’s a really poor show for Physics. This polarising trend is not surprising. People like to be where they feel they belong. If you’re conspicuously outnumbered, you’re more likely to feel uncomfortable. Data show that girls are more likely to continue with Physics in all-girls schools. Also, as we’ve seen, there seems to be a clustering of boys’ subjects and girls’ subjects, and developing these reputations can make it difficult for people to go against stereotypes. This impacts both how people view themselves and others, potentially impacting perceived competence (e.g., for Physics, Gonslaves 2014a, 2014b). These cultural biases are something we need to work against if we’re going the get the best mix of students (I guess it’s good we have all these Psychologists and Sociologists to help figure this out).

I’d recommend trying the excellent (and adorable) Parable of the Polygons to see how biases can become magnified.


At A level, some subjects are favoured by boys or by girls. This imbalance gets larger during the transition from AS to A2. Physics is one of the most popular subjects at AS level, but lags behind the other main sciences. It has a poor gender ratio, which notably gets worse going from AS to A2. Physics is (arguably) the the most awesome subject, so we should do more to show that is for everyone. If you’d like to play around the data (and don’t fancy typing it out yourself), I have it available via Google Drive.

(For disclosure: I took Geography at AS, and Physics, Maths and Further maths at A2).

Equation etiquette

Mathematics can be beautiful. Equations are an important component of technical writing, but getting their presentation correct can be tricky. There are many rules about their formatting, and these can seem somewhat arbitrary. Just like starting with the outermost knife and fork at a fancy dinner, or passing the port to the left, these can seem rather ridiculous when you first learn them, but there is some logic to them. Here, I give a short guide to the proper etiquette of including equations in your writing.

0 Make introductions

The simplest rule: explain what your symbols mean. The dinner-party equivalent would be to introduce your guests, so that everyone knows whom they have to attempt conversation with. For an equation to be of any use, people need to know what it means. This can be especially important as some symbols are commonly used for different quantities. Introduce your readers to your symbols promptly, so that the equation makes sense. For example,

“Ohm’s law says that the voltage across a resistor is

V = IR,

where I is the current flowing through the resistor and R is the resistance of the resistor.”

Here, I left the definition of V implicit, but hopefully everyone’s now acquainted, so we can chat (probably about electronics) until the soup is ready.

Depending on your audience, there are some things you can get away without introducing. The mathematical constant \pi is always referred to as pi, so you can usually skip the definition of it being the ratio of a circle’s circumference to its diameter. \pi is the superstar guest that needs no introduction. If you are using the symbol for something else, make sure to make that clear!

Pi pie!

Pi pie! Perfect for any mathematical dinner party. Technically, there’s 2\pi of pie here. Credit: Tasty Retreat

While not as famous as \pi, the mathematical constants e, the base of the natural logarithm, and i = \sqrt{-1}, the imaginary unit, can sometimes be left undefined. They are dinner-party regulars, so as long as your guests have been invited along a few times before, they should have met. Unlike \pi, e and i are frequently used for other quantities, so if there’s chance of there being some confusion, play it safe and make the introduction (remember, no-one like having to ask the names of people that they’ve met before).

Finally, some of the fundamental physical constants like the speed of light c, the Newtonian constant G, Boltzmann’s constant k and the reduced Planck constant \hbar, can sometimes be left unintroduced if writing for professional physicists. They are guests that went to university together, so you can assume they know each other. If there is any chance of confusion though, make sure to introduce them. Try to never use a symbol for any of the constants that is not their usual one, that’s like giving a guest a new nickname for the purpose of the party. It will lead to all sorts of confusion, which might be amusing in a sit-com, but less so in scientific writing

Never use the same symbol for two different quantities. Just like having a seating plan with two identical names, this leads to confusion, arguments over who gets to sit next to the awesome physicist, and people being stabbed with forks. Using subscripts or superscripts, or a different font are common ways of avoiding a clash.

1 Punctuate properly

Equations should form a central component of your text. They are part of your sentences. Accordingly, they should be punctuated properly so that they make sense. This is like chewing with your mouth closed: no-one likes to see a mess.

It can be hard to put equations into words, to figure out where to put punctuation. However, they can usually be read as “left-hand side equals right-hand side”. Here, “equals” is a verb. Often an equation will need to be followed by comma, as above. Missing out punctuation is especially obvious when the equation comes at the end of a sentence and there’s no full stop.

Starting a sentence with an equation is a little weird, like serving the sweet before the soup, but I don’t think there’s anything to stop you. Consider the following examples.

“The most famous equation in physics is E = mc^2. This explains the equivalence of energy E and mass m, converting using the speed of light c.”

E =mc^2 is the most famous equation in physics. Here, E is the energy equivalent of mass m, and c is the speed of light in a vacuum.”

2 Fonts, roman, italic

Lend me you ears, I come with some of the finer details, like which fork to use. Variables are typeset in italics. This makes it easy to spot with letters are mathematical quantities and which are just plain text: a is a variable and a is just a short word.

Not everything that appears in an equation should be italicised. Numbers; operators like +, - and \times, and brackets (\ldots) are left as they are. These are always just themselves, so there’s no need to italicise, they are left roman (upright).

Function names, when more than one letter, are not italicised. For example \sin, \log or \min. This lets you know that these letters can’t be broken up, they come as a single unit. For example

\displaystyle \frac{sinx}{cosx} = \frac{in}{co},


\displaystyle \frac{\sin x}{\cos x} = \tan x.

Related to this, is the question of whether you should italicised the differential \mathrm{d}? I like to have it roman so it’s

\displaystyle \frac{\mathrm{d}x}{\mathrm{d}y} \quad and \quad \int f(x)\, \mathrm{d} x.

I think this makes it clear that the infinitesimal element \mathrm{d}x can’t be broken up (you can’t cancel \mathrm{d}). However, this is not universal, so I think this is much like whether you should prod or crush the peas onto your fork.

Subscripts and superscripts often lead to confusion. If they are part of a variable’s name, should they always be italicised? The answer is no: they should be treated as if they were in the main text. If I want to specify the area of a circle, it would be A_\mathrm{circle}, as circle is just a regular word. If I want to specify the coordinates of point \mathrm{P}, they are (x_\mathrm{P},\,y_\mathrm{P}), as \mathrm{P} is the name of the point, not a variable. If I wanted to talk about heat capacity, then the heat capacity at constant volume is C_V and the heat capacity at constant magnetic flux density is C_B because I’m using V and B to specify the volume and magnetic field respectively.

All this seems to make sense to me. It might seem strange that there’s a specific item of cutlery for each course, but it is easier to cut a steak with a steak knife than a butter knife, so there may be some logic to it. Similarly, the typesetting of maths does convey some meaning.

Sadly, there is a common exception to the rule, upper-case Greek letters are often not italicised, but are left upright, e.g. \Theta. (Lower-case Greek letters are italicised, as are our Latin upper-case letters). It could be that this gives a way of distinguishing between an upper-case beta \mathrm{B} and a capital B, chi \mathrm{X} and X, etc. However,  I think this is just because they look odd in some fonts. Italicising them wouldn’t be wrong. (Although, the summation symbol \sum and product symbol \prod are operators, and so should never be italicised).

3 Laying out units

Forgetting to include units is much like forgetting your trousers at a dinner party. It’s a definite faux pas, not to mention painful if you drop some of that hot soup. However, unlike the wearing of trousers, there is an international guideline on how to correctly use units. Units appear after a number separated by a small non-breaking space, e.g. x = 2.3~\mathrm{m}. The space needs to be non-breaking so that it’s never separated from the number, which would be painful.

Groundskeeper Willie

Trousers are not standardised, but units are! The Springfield Police are shocked when Willie forgets his. Credit: Fox

You may have noticed that units are not italicised. This makes them readily identifiable, and also avoids any confusion that a millimetre is the same as a square metre or that one hertz per henry could be z. Not italicising units means there’s a clear difference between T = 5~\mathrm{s} and T = 5s. The first indicates a time of five seconds, the second that T is five times s, whatever that might be. We can also write things like s = 5~\mathrm{s} without them being nonsense.

When making compound units, use negative powers rather than a slash so there are no ambiguities. It’s difficult to figure out \mathrm{m/s^2/kg^3}, but \mathrm{m~s^{-2}~kg^{-3}} is clear. You don’t want everyone pondering if you’ve accidentally put your trousers on back-to-front.

Finally, when plotting graphs, units should be included in the axis labels. I like to think of graphs just being of pure, dimensionless numbers, hence I need to divide out the units, e.g. T/~\mathrm{s} for time in seconds or C_V/(\mathrm{J~K^{-1}}) for heat capacity.

4 Use the right symbol for the job

Trying to eat your soup with your crab fork is not going to end well. You should always use the right tool for the job. When writing maths, this means using the correct symbol. The multiplication sign \times is not an x, and the minus sign - is not a hyphen.

5 Close your brackets

No parsing scripts should be harming in the reading of this

Pure evil. Credit: xkcd

To close, some tips on brackets. Brackets should always come in an (equally-size) pair. They should be large enough to enclose their contents. When eating, you should cut your food up into bite-size pieces, you can’t chop up equations in the same way, so instead you resize the brackets.

When nested brackets, use different types of brackets so it’s clear which term ends where. It’s usual to start with parentheses (\ldots), then use square brackets [\ldots], and then braces \{\ldots\}. Unlike with cutlery, you start inside and work your ways out. For example, making something up,

\displaystyle \exp\left\{-(1 + 2\xi)\left[(\xi - 1)^2 + \cos \left(\frac{\pi \xi}{2}\right)\right]^{-1/2}\right\}.

If you need more than three levels, you usually cycle round again.

There are a few cases where a particular type of bracket is used. Angle brackets \langle\ldots\rangle are often used for an average. Square brackets are often used to enclose the argument of a functional. Curly braces are often used for limits, \lim_{x\,\rightarrow\,0} \{\mathrm{sinc}\,x\} = 1, or Fourier transforms, \mathscr{F}_k\{f(x)\} = \tilde{f}(k). The important thing is to be clear, to make it easy for the reader to distinguish which brackets matches to which other.

That brings us to the end. We’ve closed all our brackets, and put our knife and fork together on our plate. Presenting equations clearly, like writing clearly, makes writing easy to understand. Paying attention to the details, making sure that you dot all your is and cross all your \hbars, creates a good impression, it shows you’re careful and that you care about your work. You may even get invited out to dinner again.

Tips for scientific writing

Second year physics undergraduates at the University of Birmingham have to write an essay as part of their course. As a tutor, it is my job to give them advice on how to write in a scientific style (and then mark the results). I have assembled these tips to try to aid them (and make my marking less painful).

Writing well is difficult. It requires practice. It is an important skill, yet it is something that I do not believe is frequently formally taught (at least in the sciences). However, there are many resources online, and many universities offer support for learning key skills, like the University of Birmingham Library. Scientific and other technical writing can be especially hard, as it has its own rules that can be at odds with what we learn at school (when studying literature or creative writing). Reading the work of others is a good way for figuring out what works well and what does not.

In this post, I include some tips that I hope are useful (not everyone will agree). I begin by considering how to plan and structure a piece of writing (section 1), from the largest scale (section 1.2) progressing down to the smallest (section 1.4); then I discuss various aspects of technical writing (section 2), both in terms of content and style, including referencing (section 2.5), which is often problematic, and I conclude with some general editing advice (section 3) before summarising (section 4). If you have anything extra to add, please do so in the comments.

1 Structure and planning

The structure of your writing is important as it reflects the logical flow of your arguments. It is worth spending some time before you start writing considering what you want to say, and what is the best order for your ideas. (This is also true in exams: I have found when trying to answer essay questions it is worth the time to spend a couple of minutes planning, otherwise I am liable to miss out an important point). I frequently get frustrated that I must write linearly, one idea after another, and cannot introduce multiple strands at a time, with arguments intertwining with each other. However, putting in the effort to construct a clear progression does help your reader.

1.1 Title and audience

The first thing to consider is what you want to write and who is going to read it. Always write for your audience, and remember that professional scientists and the general public look for different things (this blog may be a poor example of this, as different posts are targeted towards different audiences).

Having thought about what you want to say, pick a title that reflects this. Don’t have a title “The life and works of Albert Einstein” if you are only going to cover special relativity, and don’t have a title “Equilibrium thermodynamics of non-oxide perovskite superconductors” if you are writing for a general audience. If your title is a question, make sure you answer it. It might be a good idea to write your title after you have finished your main text so that you can match it to what you have actually written.

1.2 Beginning, middle and end

To help your audience understand what you are telling them, begin with an introduction, and end with a summary. This is also true when giving a talk. Start by explaining what you will tell them, then tell them, then tell them what you told them. Repetition of key ideas makes them more memorable and help to emphasise what your audience should take away.

At the beginning, introduce the key ideas you will talk about. If you are writing an essay titled “The Solar Neutrino Problem“, you should explain what a solar neutrino is and why there is a problem. You might also like to explain why the reader should care. Sketching out the contents of the rest of the work is useful as it prepares the reader for what will follow: it’s like warm-up stretches for the mind. The introduction sets the scene for the arguments to follow.

The main body of your text contains most of the information, this is where you introduce your ideas and explain them. It is the burger between the buns of the introduction and conclusion. For longer documents, or subjects with many aspects, you might consider breaking this up into sections (and subsections). Using headings (perhaps numbered for reference) is good: skimming section headings should give an outline of the contents. Some sections within the main body might be sufficiently involved to merit their own introduction and summary. There should be a clear progression of ideas: if you find there is a big jump, try writing some text to cover the transition (“Having explained how neutrinos are produced in the Sun, we now consider how they are detected on the Earth”).

After presenting your arguments, it is good to summarise. As an example, a summary on the solar neutrino problem could be:

“Experiments measuring neutrinos from the Sun only detected about a third as many as expected. This could indicate either a problem with our understanding of solar physics or of particle physics. It is not possible to modify solar models to match both the measured neutrino flux and observations of luminosity and composition; however, the reduced flux could be explained by introducing neutrino oscillations. These were subsequently observed in several experiments. The solar neutrino problem has therefore been resolved by introducing new particle physics.”

Don’t introduce new arguments at this stage, this is just as unsatisfying as reading a murder mystery and discovering the murderer was someone never mentioned before. In my solar neutrino example, both the solar models and neutrino oscillations should have been discussed. Distilling your argument down to few lines also helps you to double-check your logic.

Either as part of your summary, of following on from it, end your writing with a conclusion. This is what you want your audience to have learnt (it should be the answer to your question). It is OK if you cannot produce a concrete answer, there are many cases where there is no clear-cut solution, perhaps more data is needed: in these cases, your conclusion is that there is no simple answer. To check that you have successfully wrapped things up, try reading just your introduction and conclusion; these should pair up to form a delicious (but bite-sized) sandwich.

1.3 Paragraphs

On a smaller scale, your writing is organised using paragraphing. Paragraphs are the building blocks of your arguments; each paragraph should address a single point or idea. Big blocks of text are hard to read (and look intimidating), so it is good to break them up. You can think of each paragraph as a micro-essay: the first sentence (usually) introduces the subject, you then go on to elaborate, before reaching a conclusion at the end (see section 1.2). To check that your paragraph sticks to a single point (and doesn’t need to be broken up), try reading the first and last sentences, usually they should make sense together.

1.4 Sentences

Paragraphs are constructed from sentences. Ensure your sentences make sense, that they are grammatically correct and that their subject is clear.

Vary your sentence length. In technical writing there is often the temptation, even amongst the best writers, to include long, convoluted sentences in order to fully describe a complicated idea and include all the relevant details, but these can be hard to read, both because of the complexity of their structure, which may require significant mental effort to unpack, and because by the time they finally conclude, the reader has forgotten the initial topic of the over-long, rambling sentence. Brevity gives impact. Shorter sentences are easier to understand. Breaking up your ideas helps the reader. Short sentences also get boring. They seem repetitive. They are tiring to read. They can send your reader to sleep. It is, therefore, better to have a range of sentence lengths. Include some short. In addition to these, have some longer sentences, as these allow you to join up your ideas. If you are unsure where to break up long sentences, look for commas (or semi-colons, etc.); if you are unsure where to put commas, read the sentence and see where you would pause.

2 Writing style and referencing

Having discussed how to structure your writing, we now move on to what to write. Technical writing has some specific requirements with regards to content, these might seem peculiar when first encountered. I’ll try to explain why we do certain things in technical writing, and give some ideas on how to incorporate these ideas to improve your own writing.

2.1 Be specific

The most common mistake I come across in my students’ work is the failure to be specific. The following two points (sections 2.2 and 2.3) are closely related to this. As an example, consider making a comparison:

  • Poor — “Nuclear power provides more energy than fossil fuel.”
  • Better — “Per unit mass of fuel, nuclear fission releases more energy than the burning of fossil fuel.”
  • Even better — “Nuclear fission can produce ~8000 times as much energy per unit mass of fuel as burning fossil fuels: the same amount of energy is produced from 16 kg of fossil fuels as by using 2 g of uranium in a standard reactor (MacKay, 2008).”

Here, we have specified exactly what we are comparing, given figures to allow a quantitative comparison, and provided references for those figures (see section 2.5). If possible, give numbers; don’t say “many ” or “lots” or “some”, but say “70%”, “9 billion” or “six Olympic swimming pools”.

Weak modifiers like “very”, “quite”, “somewhat” or “highly” are another example where it is better to be specific. What is the difference between being “hot” and being “very hot”? I might say that my bowl of soup is very hot, but does that tell you any more than if I just said it was hot? It is tempting to use these words for emphasis, surely if I were talking about the surface of the Sun we can agree that’s very hot? Not if you were to compare it to the centre of the Sun! Often, what is hot or cold, big or small, fast or slow depends upon the context. What is hot for soup is cold for the Sun, and what is cold for soup is hot for superconductors. It is much better to make distinctions by using figures: “The surface of the Sun is about 6000 K”.

It is OK to use “very” if you define the range where this is applicable, for example “High frequency radio waves are between 3 MHz and 30 MHz, very high frequency radio waves are between 30 MHz and 300 MHz ultra high frequency radio waves are between 300 MHz and 3 GHz.”

2.2 Provide justification

When putting forward an argument, it is necessary to include some evidence or justification to back it up. It is not sufficient merely to assert your opinion because you need the reader to follow your reasoning. If you are using someone else’s argument, you should provide a citation (section 2.5); the reader can then check there to find the reasoning. However, if it is an important point you might like to add some exposition. If you are being good about providing quantitative statements (section 2.1), you are already part way there as you can use those figures a back-up. For example, if discussing global warming, it is easy to argue it is important if you have already included figures on how many people would lose their homes to rising sea-levels, or if comparing materials, it is straightforward to argue that aluminium is better for making aeroplanes than steel if you have already included their densities. Sometimes, all that is required is an explanation of your reasoning, for example, “It is a good idea to build nuclear power plants because this reduces reliance on fossil fuels” or “It is not advisable to lick the surface of the Sun because it doesn’t taste of golden syrup.” Here, the reader might disagree that it is a good idea to build nuclear power plants, but they understand that you are using dependence on fossil fuels as an argument instead of, say, environmental issues, or the reader might agree that it is a bad idea to lick the Sun, but might have been thinking more about its temperature than its flavour. Even if the reader does not agree with your conclusions, they should understand how you reached them.

A similar idea is to show rather than tell. Don’t tell me that something is a fascinating topic or an exciting concept, get on with explaining it! Similarly, don’t just say something is important, but explain why it is important. This allows the reader to decide upon things themselves, if you have justified your arguments then they should follow your logic.

2.3 Use the correct word

In technical writing there is often a specific word that should be used in a particular context. In common usage we might use weight and mass interchangeably, in physics they have different meanings. This sometimes trips people up as they naturally try to find synonyms to reduce the monotony of their work. Always use the correct term.

Technical language can be full of jargon. This makes things difficult to understand for an outsider. It is important to define unfamiliar terms to help the reader. In particular, acronyms must be defined the first time they are used. As an example, “When talking about online materials, the uniform resource locator (URL), otherwise known as the web address, is a string of characters that identifies a resource.” Avoid jargon as much as possible; try to always use the simplest word for the job. It will be necessary to use technical terms to describe things accurately, but if they are introduced carefully, these need not confuse the reader.

A particular pet-peeve of mine is the use of scare quotes, which I always read as if the author is making air quotes. If quoting someone else’s choice of phrase then quotation marks are appropriate, and a reference must be provided (section 2.5). Most of the time, these quotation marks are used to indicate that the author thinks the terminology isn’t quite right. If the terminology is incorrect, use a different word (the correct one); if the terminology is correct (if that is what is used in the field), then the quotation marks aren’t needed!

2.4 Use equations and diagrams

Most physics problems involve solving an equation or two. For these mathematical questions, I am always encouraging my students to explain their work, to use words. When writing essays, I find they have the opposite problem: they only use prose and don’t include equations (or diagrams). Equations are useful for concisely and precisely explaining relationships, it is good to include them in writing.

Equations may put off general readers, but they improve the readability of technical work. Consider describing the kinetic energy of a (non-relativistic) particle:

  • With only words — “The kinetic energy of a particle depends upon its mass and speed: it is directly proportional to the mass and increases with the square of the speed.”
  • Using an equation — “The kinetic energy of a particle E is given by E = (1/2)mv^2, where m is its mass and v is its velocity.”

The second method is more straightforward, there is no ambiguity in our description, and we also get the factor of a half so the reader can go away at calculate things for themselves. This was just a simple equation; if we were considering something more complicated, such as the kinetic energy of a relativistic particle

\displaystyle E = \left(\frac{1}{\sqrt{1- v^2/c^2}} - 1\right)mc^2,

where c is the speed of light, it is much harder to produce a comprehensive description using only words. In this case, it is tempting to miss out reference to the equation. Sometimes this is justified: if the equation is too complicated a reader will not understand its meaning, but, in many cases, an equation allows you to show exactly how a system changes, and this is extremely valuable.

When including an equation, always define the symbols that you are using. Some common constants, such as \pi, might be understood, but it is better safe than sorry.

Equations should be correctly punctuated. They are read as part of the surrounding text, with the equals sign read as the verb “equals”, etc.

Using diagrams is another way of providing information in a clear, concise format. Like equations, diagrams can replace long and potentially confusing sections of text. Diagrams can be pictures of experimental set-up, schematics of the system under discussion, or show more abstract information, such as illustrating processes (perhaps as a flow chart). The cliché is that a picture is worth a thousand words; as diagrams are so awesome for conveying information, I’m not even going to attempt to give an example where I try to use only words. Below is as example figure, which I have chosen as it also includes equations.

“Figure 1 shows the proton–proton (pp) chain, the series of thermonuclear reactions that provides most (~99%) of Sun’s energy (Bahcall, Serenelli & Basu, 2005). There are several neutrino-producing reactions.”

The pp chain

Figure 1: The thermonuclear reactions of the pp chain. The traditional names of the produced neutrinos are given in bold and the branch names are given in parentheses. Percentages indicate branching fractions. Adapted from Giunti & Kin (2007).

Graphs can be used to show relationships between quantities, or collections of data. They can be used for theoretical models or experimental results. In the example below I show both. Graphs might be useful for plotting especially complicated functions, where the equation isn’t easy to understand. There are many types of graph (scatter plots, histograms, pie charts), and picking the best way to show your data can be as challenging as obtaining it in the first place!

“In figure 2 we plot the orbital decay of the Hulse–Taylor binary pulsar, indicated by the shift in periastron time (the point in the orbit where the stars are closest together). The data are in excellent agreement with the prediction assuming that the orbit evolves because of the emission of gravitational waves.”

Periastron shift of binary pulsar

Figure 2: The cumulative shift of periastron time as a function of time of the Hulse–Taylor binary pulsar (PSR B1913+16). The points are measured values, while the curve is the theoretical prediction assuming gravitational-wave emission. Taken from Weisberg & Taylor (2005).

All diagrams should have a descriptive caption. It is usually good to number these for ease of reference. If you are using someone else’s figure, make sure to cite them in the caption (see section 2.5).

Tables can also be used to present data. Tables might be better than plots for when there are only a few numbers to present. Like figures, tables should have a caption (which includes relevant references if the data is taken from another source), they should be numbered, and they should be referred to explicitly in the text.

When writing, it is useful to remember that different people learn better through different means: some prefer words, some love equations, and other like visual representations. Including equations and figures can help you communicate effectively with a wider audience.

There are conventions for how to present equations, graphs and tables. I shall return to this in future posts. The rules may seem arcane, but they are designed to make communication clear.

2.5 Referencing

At the end of any good piece of technical writing there should be a list of references, hence I have tackled referencing last in the section. (Sometimes this is done in footnotes rather than the end, but I’m ignoring that). However, referencing should not be considered something that is just done at the end, or something that is tacked on at the end as an after-thought; it is one of the most important components of academic writing.

We include references for several reasons:

  1. To show the source of facts, figures and ideas. This allows readers to verify things that we quote, to double-check we’ve not made an error or misinterpreted things. It also shows distinguishes what is our own from what we have taken from elsewhere. This is important in avoiding plagiarism, as we acknowledge when we use someone else’s work.
  2. To provide the reader with a further source of information. It is not possible to explain everything, and a reader might be interested in finding out more about a topic, how a particular quantity was measured or how a particular calculation was done. By providing a reference we give the reader something further they can read if they want to (that doesn’t mean our work shouldn’t make sense on it’s own: you should be able to watch The Avengers without having seen Iron Man, but it’s still useful to know what to watch to find out the back-story). By following references readers can see how ideas have developed and changed, and gain a fuller understanding of a topic.
  3. To give credit for useful work. This is linked to the idea of not claiming the ideas as your own (avoiding plagiarism), but in addition to that, by referencing something you are publicising it, by using it you are claiming that it is of good-enough quality to be trusted. If you are to look at an academic article you will often see a link to citing articles. The number of citations is used as a crude measure of the value of that paper. Furthermore, this linking can allow a reader to work forwards, finding new ideas built upon those in that paper, just as they can work backwards by following references.
  4. To show you know your stuff. This might sound rather cynical, but it is important to do your research. To understand a topic you need to know what work has been done in that area (you can’t always derive everything from first principles yourself), and you demonstrate your familiarity with a field by include references.

You must always include citations in the text at the relevant point: if you use an idea include your source, if you introduce a concept say where it came from. It is not acceptable just to have a list of references at the end: does the reader have to go through all of these to figure out what came from where?

There are multiple styles for putting citations in text. The two most common are the following:

  • Numeric (or Vancouver) — using a number, e.g., [1], where the references at the end form an ordered list. This has the advantage of not taking up much space, especially when including citations to multiple papers, e.g. [1–5].
  • Author–year (or Havard) — using the authors and year of publication to identify the paper, e.g., (Einstein, 1905). This has the advantage of making it easier to identify a paper: I’ve no idea what [13] is until I flick to the end, but I know what (Hulse & Taylor, 1975) is about.

Which style you use might be specified for you or it might be a free choice. Whichever style you use, the important thing is to include relevant references at the appropriate place in the text.

Having figured out why we should reference, where we should put references and how to include citations in the text, the last piece is how to assemble the bibliographic information to include at the end (or in footnotes). Exactly what information is included and how it is formatted depends on the particular style: there are endless combinations. Again, this might be specified for you or might be a free choice, just make sure you are consistent. Basic information that is always included are an author (this may be an organisation rather than a person), so we know who to attribute the work to, and a date so we know how up-to-date it is. Other information that is included depends upon the source we are referencing: a journal article will need the name of the journal, the volume and page number; a book will need a title, edition and publisher; a website will need a title and URL, etc. We need to include all the necessary information for the reader to find the exact source we used (hence we need to include the edition of a book, the date updated or written for a website, and so on).

There are numerous guides online for how to format references correctly. Some software does it automatically (I use Mendeley to produce BibTeX, but that’s not for everyone). The University of Birmingham has a guide to using Havard-style referencing that is comprehensive.

A final issue remains of which sources to reference: how do you know that a source is reliable? This is an in-depth question, so I shall return to it is a dedicated post.

3 Editing

Writing isn’t finished as soon as you have all your ideas on the page, things often take some polishing up. Some people like to perfect things as they go along, others prefer to get everything down in whatever form and go back through after. Here, I conclude with some tips for editing.

3.1 Be merciless

Keep your writing short. Don’t waste your readers’ time or overcomplicate things. Cut unnecessary words.

There are some phrases that are typically superfluous:

  • “Obviously…” — If it is obvious, then the reader will realise it; if it’s not, you are patronising them.
  • “It should be noted that…” — That would be why it’s written down! (I hope you are not writing things that shouldn’t be noted).
  • “Remember that…” — You’re reminding the reader by writing it.
  • Any of the modifiers like “very”, “quite” or “extremely” mentioned in the section 2.3.

3.2 Proof-read

The single best method to improve a piece of writing is to proof-read it. Reread what you have written to check that it says what you think it should. I find I have to wait for a while after writing something to read it properly, otherwise I read what I intended to write rather than what I actually did. Having others read it is an excellent way to check it makes sense (especially if you are not a native English speaker); this is best if they are representative of your target audience.

I hate it when others find a mistake in my writing. It’s like rubbing a cat the wrong way. However, each mistake you find and correct makes your writing a little better, and that’s really the important thing.

4 Summary

In conclusion, my main tips for good scientific writing are:

  • Plan what you want to tell your audience and how they will take your message away.
  • Say what you’re going to say (introduction), then say it (main text), then say what you said (conclusion).
  • Have a clear, logical flow, with one point per paragraph.
  • Be specific and back up with your points with quantitative data and references.
  • Use equations and diagrams to help explain.
  • Be concise.
  • Proof-read (and get a second opinion).

If you have any further ideas for improving essay writing, please leave a comment.

White lab coats, pink tutus and camouflage fatigues

In this post I contemplate the effects of stereotypes and biases. I hope that this will encourage you to examine these ideas too. I promise I’ll get back to more science soon.

Just over a week ago, I helped with an outreach event for year nine students. Some of the astrophysics PhD students and I ran an interactive lecture on gravity and its importance in astrophysics. These type of events are fun: you get to teach some physics to a (usually) enthusiastic audience, and hopefully inspire them to consider studying the subject. I also get to play with our Lycra Universe. I think it’s especially important to show students what a university environment is like and have them interact with real scientists. It is important to counter the stereotype that studying science means that you’ll spend all day in a lab wearing a white lab coat. (Although that would be cool. I’d want goggles too, and maybe a doomsday device).

This event was to promote the studying of STEM subjects. That’s science, technology, engineering and mathematics, because there’s nothing like an acronym to make things accessible. It is often argued that we need more people trained in STEM subjects for the economy, industry, or just so we can finally get pizza over the Internet. I like to encourage people to study these areas as I think it’s good to have a scientifically-literate population. Also, because science is awesome! The event was aimed specifically at encouraging a group who are under-represented at university-level STEM, namely girls.

There has been much written on gender and subject choice. I would recommend the Closing Doors report by the Institute of Physics. I will not attempt to unravel this subject. In all my experience, I have never noticed any difference in aptitude between genders. I don’t believe that the ability to pee standing up gives any advantage when studying physics—one could argue for a better understanding of parabolic motion, but anyone who has paid attention to the floor in the gents (I advise against this), knows this is demonstrably not the case. I assume the dominant factors are social pressures: a vicious circle of a subject becoming more associated with one gender, which makes people feel self-conscious or out of place studying it. Also: there are always bigots. It’s a real shame to be potentially missing out on capable scientists. There have been many attempts to try to counter this trend, to break the cycle—some of them truly awful.

Good arguments have been made that the gender segregation of toys pushes girls away from science and technology from an early age. (For some reason, there seems to be a ridiculous idea that women can only relate to things that are pink). It makes sense to me that if only boys get the chemistry sets and construction toys, then they are going to be more numerous in the STEM subjects. The fact that a few female LEGO scientists merits coverage in nation newspapers, the BBC, etc. shows something isn’t quite right.

We are all influenced by our childhoods, and this got me thinking: I know of negative impacts for women from these gender biases, what are they for men? If women are under-represented in engineering, maths and physics, then men must be under-represented somewhere else to balance things: namely English, biology (conspicuous amongst the STEM subjects) and languages. We are short of male teachers and nurses. It seems that men are pushed away from caring careers or those with emphasis on communication.

The lack of men in certain professions is a problem, although I would say less so than the continued under-representation of women at senior positions (say as professors, CEOs or members of government). I was about to relax, since I hadn’t uncovered yet another unconscious bias to add to the list. Then I checked the news. I don’t know what’s in the news when you’re reading this, but at the time it was conflict in Ukraine, Iraq and Israel–Palestine—I assume things are much better in the future? One thing that struck me was that the combatants in the photos were almost exclusively men. It then occurred to me that for every girl who plays with a ballerina doll, there is a boy who plays with an action figure with a weapon. I’m not as naive as to suggest it’s a simple as growing up to be exactly like your toys (I, regrettably, am neither a dinosaur nor a cuddly elephant), but perhaps it is worth keeping in the front of our mind what identities we associate with each gender and how we project these onto children. I don’t want to say that being a ballerina isn’t a good vocation or hobby, or that being a soldier is a bad career. (Curiously, I believe that some of the requirements to be a good ballet dancer or soldier overlap, say discipline, determination, physical fitness and, perhaps, empathy). However, I think it is dangerous if we raise girls who primarily aspire to be pretty, and boys who resolve conflict through violence (men are both more likely to be victims of homicide and suicide).

In conclusion, stereotypes can be damaging, be it that scientists are all socially-awkward comic-book geeks as in The Big Bang Theory, that men can’t talk about their feelings, or that women must be mothers. There is a balance between the genders: by assigning one quality to a particular gender, you can push the other away. Mathematical ability shouldn’t be masculine and compassion shouldn’t be feminine. This is not a new idea, but conveniently coincides with Emma Watson’s wonderful speech for the UN as part of the HeForShe campaign. Cultural biases might be more significant than you think, so give them some extra attention. Sexism hurts everyone, so let’s cut it out and all go play with some LEGO.

The Big Bang Theory

The Big Bang Theory‘s popularity has been credited with encouraging more students to take physics. The cast reflects traditional stereotypes: the men are physicists, an astronomer and an engineer, the women are two biologists and Penny.

How sport is like science

Athene Donald, Professor of Experimental Physics and soon-to-be Master of my old college, Churchill, recently blogged about how athletics resembles academia. She argued that both are hard careers: they require many years of training, and even then success is not guaranteed—not everyone will reach the top to become an Olympian or a Professor—there is a big element of luck too—a career can stall because of an injury or because of time invested in a study that eventually yields null results, and, conversely, a single big championship win or serendipitous discovery can land a comfortable position. These factors can make these career paths unappealing, but still most people who enter them do so because they love the area, and have a real talent for the field.

The Breakfast Club

As The Breakfast Club taught us, being into physics or sports can have similar pressures.

I find this analogy extremely appealing. There are many parallels. Both sports and academic careers are meritocratic and competitive. Most who enter them will not become rich—those who do, usually manage it by making use of their profile, either through product endorsement or through writing a book, say Stephen Hawking, or Michael Jordan (although he was still extremely well paid). Both fields have undisputed heavy-weights like Einstein or Muhammad Ali, and media superstars like Neil deGrasse Tyson or Anna Kournikova; both have inspirational figures who have overcome adversity, be they Jesse Owens or Emmy Noether, and idols whose personal lives you probably shouldn’t emulate, say Tiger Woods or Richard Feynman. However, I think the similarity can stretch beyond career paths.

Athene says that although she doesn’t participate in athletics, she does enjoy watching the sport. I’m sure many can empathise with that position. I think that this is similarly the case for research: many enjoy finding out about new discoveries or ideas, even though they don’t want to invest the time studying themselves. There are many excellent books and documentaries, many excellent communicators of research. (I shall be helping out at this year’s British Science Festival, which I’m sure will be packed with people keen to find out about current research.) However, there is undoubtedly more that could be done, both in terms of growing the market and improving the quality—reporting of science is notoriously bad. If you were to go into any pub in the country, I’d expect you’d be able to find someone to have an in-depth conversation with about how best to manage the national football team, despite them not being a professional footballer. Why not someone with similar opinions about research council funding? Can we make research as popular as sport?

Increasing engagement with and awareness of research is a popular subject, most research grants with have some mention of wider impact; however, I don’t think that this is the only goal. According to UK government research, many young students do enjoy science, they just don’t feel it is for them. The problem is that people think that science is too difficult. Given my previous ramblings, that’s perhaps understandable. However, that was for academic research; science is far broader than that! There are many careers outside the lab, and understanding science is useful even if that’s not your job, for example when discussing subjects like global warming or vaccination that affect us all. Coming back to our sports analogy, the situation is like children not wanting to play football because they won’t be a professional. It’s true that most people aren’t good enough to play for England (potentially including members of the current squad, depending upon who you ask in that pub), but that doesn’t mean you can’t enjoy a kick around, perhaps play for a local team at weekend, or even coach others. Playing sports regular keeps you physically fit, which is a good thing™; taking an interest in science (or language or literature or etceteras) keeps you mentally fit, also a good thing™.

Chocolate models

Chocolate is also a good thing™. However, neither Nobel Prizes nor Olympic Medals are made of chocolate, something I’m not sure that everyone appreciates. I’d make the gold Olympic models out of milk chocolate, silver out of white and bronze out of dark. The Nobel Prize for Medicine should contain nuts as an incentive to cure allergies; the Prize for Economics should be mint(ed) chocolate, the Peace Prize Swiss chocolate, the Chemistry Prize should contain popping candy, and the Physics Prize should be orange chocolate (that’s my favourite).

How to encourage more people to engage in science is a complicated problem. There’s no single solution, but it is something to work on. I would definitely prefer to live in a science-literate society. Stressing applications of science beyond pure research might be one avenue. I would also like to emphasis that it’s OK to find science (and maths) hard. Problem solving is difficult, like long-distance running, but if you practise, it does get easier. I can only vouch for one side of that simile from personal experience, but since I’m a theoretician, I’m happy enough to state that without direct experimental confirmation. I guess that means I should take my own advice and participate more myself: spend a little more time being physically active? Motivating myself is also a difficult problem.