Maths anxiety, disaffect, or just risk aversion?
I very much enjoyed the event run on 1 June 2016 by the MMU QStep team and the Sigma Network on reducing mathematical anxiety in higher education and particularly among non-Maths students. This is something I’ve thought about over the past couple of years. I teach an introductory unit at Bristol in quantitative methods to undergraduates from Childhood Studies, Politics, Social policy and Sociology — both students who chose the unit on a standalone basis, and those following one of the ‘with Quantitative Methods’ degree programmes. The unit name is ‘Principles of Quantitative Social Science’: like the ‘with Quantitative Methods’ programme tag, it’s hardly trying to camouflage its content. And yet a few students are still surprised at the level of numeracy expected.
PQSS is very much intuitive — to the point that a couple of students who have done taken Maths to 18 have requested more use of formal notation. It’s not really mathematical, and yet of course there is some arithmetic, an expectation that they remember the equation for a straight line, and so on. Students who think they have made a mistake and are sticking with it anyway need the kind of support that was discussed at the MMU/Sigma event; and yet this has to be balanced against the needs of other, different, students.
Mixed Ability Groups
I was struck by Mark Brown’s point that one challenge involves working with a very large group of students of mixed ability within the curricular model of lectures and labs. This is also something I’ve found at Bristol, albeit with a smaller group: the variation in starting points (though not raw ability) is larger than for substantive units.
A small number of students fit the model described at the Sigma/MMU event of people who have spent a long time away from school Maths, who were not taught particularly well in the first place, and who have a high dose of imposter syndrome. At the same time, in social science quants teaching, we also have to cater to those who have done A-level Maths or more. Others consider that Maths is not their strong point but nevertheless carried on to 18 because it was required in their educational system. Others still are very strong, but are less engaged because they just find it less interesting than substantive modules.
We have to compete with other modules which are viscerally-enticing. Some students are concerned that they may spoil a clean sheet of 2.1 or better scores; they know where they stand with essays and perceive QM as risky. Following a QM unit also requires consistent engagement — and those who have caring responsibilities or poor health need additional support to catch up when they miss particular weeks, which is less of an issue with less linear humanities or social science modules.
Another point which was viewed positively was that good Maths doesn’t have to be done quickly. This is very reassuring for me personally, but some (not all) students are under time pressure throughout the term and devoting significantly more time to one unit over another — when the credit is ultimately the same — is a challenge. They get on with it, but are often just looking for acknowledgement that they are making this additional commitment. If the balance were wrong, they would grow resentful.
In the introductory quants modules, those who are mathematically very strong can fly. I’ve nevertheless found that they can still have conceptual sticking points. And despite finding the material easy, they’re kept busy enough with learning the mechanics of SPSS and applied social science questions regarding operationalisation and so on. Diversity and mixed ability doesn’t always mean a spectrum from weak to strong — the mathematically strong can struggle over quite surprising things.
Another interesting feature is that those who are highly able in this module have not necessarily scored highly across their degree programme as a whole, at least before that point. For those who don’t necessarily shine as much in the substantive or essay-based units, it’s very rewarding for them to find material that clicks. I’m aware of a couple of cases where early success in quants has directed students to specialise explicitly in more empirical modules, even outside the requirements of the QStep degree pathways. They figure out their strengths and play to them. It can be surprising to us, though, that students who appear to be very competent have a perception of themselves as lacking general ability, presumably on the basis of school and first year performance . This can often be a question of being a slightly late developer, or of not having found their niche yet.
Accordingly, the discussion by Sue Johnston-Wilder at the event of how to encourage a growth mindset and teach people to be more self-reliant in their learning was particularly interesting. One issue I’ve found is that my main recourse has been to meet students individually — so that they are given the chance to explain their personal story and be reassured that they will be okay — and this is harder to do for everyone as the class gets bigger, and teaching time more compressed.
Student mental well-being is a pressing issue. There appear to be clear generational effects on social and educational anxiety, with no easy or immediate answers. The need for additional and individualised support can lead us to batten down the hatches, even if only mentally.
Quite often, I read an email which requests in panic a piece of information which the student could otherwise find from MathsisFun.com in seconds. I’ve come to learn that they already know they can Google it, thanks, but are primarily looking for a signal that we’re interested in their progress and care about how they are feeling about the course as a whole. Responding positively and expansively as a matter of policy has helped me save mental energy on triaging student emails.
The organisational psychologist Adam Grant has done a lot of research to show that helping helps the helper. My personal experience is that he’s right: it’s quicker to just help (and thereby show them how to help themselves), rather than explain why it’s better in principle that they figure out how to do it themselves. Counterintuitively, a dose of helping upfront builds trust so that students feel more able to figure it out next time.
Resilience can indeed be acquired — I was lucky enough to co-supervise an excellent PhD study on this. It’s partly built through the experience of making mistakes and learning then that error is usually not fatal. Accordingly, I discuss error a lot, and not only in the statistical sense. I explain that we’re not too concerned about trivial and correctable errors that everyone makes, because these can generally be prevented by better data management and following data analysis checklists.
This is important because many students have strong discomfort at being wrong, and have selected courses from GCSE onwards which are inherently more subjective. This can nevertheless be turned to our advantage. QM can be very good for building resilience because solutions are more clearly right or wrong. Students can more easily see that there isn’t some moral or qualitative divide between a 49 and a 50, or 59 and 60. Highly transparent assessments also build trust. And seeing that three minor mistakes on the mid-term paper led to a loss of five marks can make progression from a 65 to a 70 seem much more possible. Much as we think of ourselves as being highly intrinsically-motivated, and that students would be better off by being more like us, marks matter, as do modes of assessment, which I’ll turn to next.
My reference to our use of assessment by exam raised eyebrows when chatting with another participant. It’s a trope that ‘in real life’ we don’t have to work under time pressure without being able to check details. But in fact we often do, and when I’ve written talks or briefing notes to deadline, they’re often better for being less overwrought.
This mode of assessment was agreed before I took the unit on, and I probably wouldn’t have chosen it; and yet I’ve become a firm fan. Last year, there was an open book element, where students had to study Mahalia Jackman’s recent article on infidelity attitudes in advance. In the exam, they had to give a critical evaluation of the paper’s quantitative analysis with reference to some specific technical questions. Having read the paper in advance helped assuage some anxieties regarding the exam being unseen.
Students also need it to be explained that exams aren’t intended to catch people out on trivial questions, and that we look to give credit where it’s due. One of the main virtues is that it limits the time spent on presentation and perfectionist but ineffective editing which we can occasionally see with assessed coursework.
I was one of the invigilators for the PQSS exam this year, and thought that sitting the exam itself wasn’t a bad experience for building resilience. The students could see that they were just one of hundreds in exactly the same position, and that they could only do what they could. A lot of anxiety arises from disconnection and thinking that they should be boundlessly capable. Sitting an exam together brings them face-to-face with others in the same situation, and the reality that what they submit has to end.
Sitting down for a week to revise the material also helps most students finally see how it all fits together. They can’t pick and choose their favourite essay from a list, and set aside the rest. I left the Great Hall with an armful of scripts and reassured about what we’re doing.
Motivation and Passion
I actually find that extrinsic motivation (pursuing QM for jobs and so on) is pretty high. Yet students are quite rational in prioritising assessment scores now rather than employability later. Many graduate schemes using computerised recruitment systems screen out 2.2 graduates and it seems that 2.2 degrees are perceived much less positively by some students now than when I was at university, which is unfortunate.
For those who do take QM for CV-building purposes, a fair number do not grow to love it, even after going through the experience of ‘seeing it all come together’ and getting a good grade in the exam.
I was a little surprised at first that I couldn’t convert them all. Many of us who teach it really do love it, and find it fascinating. Students can of course see this. They also notice and appreciate the overlap with substantive units, and that they are much more adept at reading empirical articles, so see that they are getting ahead in that regard. Ultimately, though, they are much more focused on getting through the programme as a whole as well as they can, which is fair enough.
Equally, there are the very good students who really could be converted, and at Bristol we’ve been lucky enough to draw some of them in over the past two years. Such students are often aware that they will do very well whatever they do; and so we have to compete for them. Those who really take to it really do find it enriching, and that it provides them with something they would not otherwise have found across the suite of substantive or theory modules.
I’ve also found it easier to persuade students who are quite keen on doing postgraduate study to switch in, because it will help them get a head-start on an MSc, and perhaps even help with Masters-level funding.
One of the answers that came out of the MMU/Sigma workshop was that stories are useful. I do use anecdotes a lot, relating to my own mistakes, and those of very established scholars. In fact, I’m half minded to put some technical mistakes back in so that I can correct them live, and so that students see that the sky doesn’t fall in if we make minor errors (a slightly risky game to play, though!)
Here are a few of the stories and metaphors I use:
· Learning QM is like learning to drive — it gives you freedom. People who are not natural drivers are expected to keep trying, so why should university work be different? Indeed, I failed my test twice when I was 30 and now do a lot of driving. Nevertheless, learning to drive younger is easier which is why undergrad is the perfect time to learn QM
· Learning QM is like strength training at the gym — it does take time, and is a different experience to writing essays (for which read cardio work) but makes you better all round; you need both types of training to be a good athlete/researcher
· This unit is like couch to 5k: some students are already at 3k, some of us still think we’re on the couch, but we’re all going to get there, just like in a ParkRun
· The introductory practicals are like following recipes when first learning to cook: once you’re fluent you’re able to select the procedures ad lib to do a beautiful, creative and tasty set of analyses of your own choice
· You are following a course essentially the same as tens of thousands of students across the world. It’s not some horrible experience devised to punish Bristol social science students in particular
· People (recruiters, admissions boards and so on) do notice when students choose difficult units. By choosing quants, they brand themselves as the type of person who has guts and takes on a harder challenge. This sets them apart already regardless of The Final Mark
… and so on.
Context and Legacy Effects
My main takeaway was that the challenges we experience are context-dependent. I very much like the idea of undergraduates working as Data Buddies — a stellar scheme run by MMU — and will explore this further. A lot, though, depends on what’s institutionally possible. I’ve already put in a request that a numeracy support hour be earmarked in the timetable, separate to the formal provision of a one-hour lecture and two-hour lab. This is because I’ve learned over the past two years that office hours are just not enough: many students work long hours part-time, or have timetable clashes, so that we spend too much time chasing slots where we’re both free. Beyond numeracy support, though, students still need to learn primarily independently; a doubling of contact time, for example, would not be sustainable.
There also seemed to be a consensus from our discussion that lectures are not the best way of teaching quants. Perhaps, but we’re in a world of second bests and chalk-and-talk is… efficient. Videoing lectures in advance to allow ‘flipping’ may help create more space in class for discussion and working through examples, so I’m planning on moving some of the lecture material online in this fashion next term. Nevertheless, it will take time and planning to do well, and we’re all time-constrained. Most innovations are incremental rather than utterly transformative in the short term, so we shouldn’t expect lecture capture or flipping to be a magic bullet.
A lot of thought needs to go into curriculum design, and to ensure that those accepted into a module or programme, having fulfilled the course prerequisites, are given the right resources to succeed. For this, departments need to think about providing wider numeracy support alongside formal tuition — just as they provide general study skills support when students first arrive at university, or academic writing support of the type provided to highly able international students. Where office hours are not enough, schemes such as the Maths Café and Data Buddies are creative and productive, offering a lot of bang for buck.
Beside this, some mathematical disaffect is not rooted in anxiety or biased perceptions, but is quite rational. Students are highly motivated by marks. There is some risk in taking a ‘different’ module, even where the risk is upside as well as downside. There are a lot of attractive modules out there. And students are increasingly time-pressured, so even where they are very open to the material, there are particular challenges in terms of catering to those who miss a few weeks, and those who take longer to get to grips with it.
Because we need to compete with other modules for students, we need to keep making sure that the effort to reward ratios are about right rather than relying only on intrinsic rewards. Some of this depends on the nature of the intake, which is not in itself a fixed thing.
Getting the model right (and scalable) is important. Passion and empathy are great, but not enough in themselves, because students are looking for more: reassurance that they will get from A to B in a reasonable time and with a fair mark.
To answer the question in the heading, Maths anxiety is something we do have to think about if teaching quants to non-scientists. But it’s not the only issue, and it’s entirely reasonable for students to just not love QM, or to feel that their choices are limited by taking it, or to ask whether it will be harder than other modules or make them less likely to get a first. Having credible answers is up to us. For those who do really get it, their experience is wholly enriched by finding they type of work at which they flourish, and this makes it wholly worthwhile.