Category Archives: Mathematics HE

Mathematical Sciences HE Curriculum Innovation Final Update

Hear me say this: Mathematical Sciences HE Curriculum Innovation Final Update (audio).

In the academic years 2010/11 and 2011/12 the Maths, Stats and OR (MSOR) Network supported a set of 32 projects on ‘Mathematical Sciences HE Curriculum Innovation’ through funding of around £250,000. This work was completed as part of the Mathematical Sciences Strand of the National HE STEM Programme.

The National HE STEM Programme was an initiative aiming to enable the HE sector to engage with schools, enhance curricula, support graduates and develop the workforce, operating through a three-year grant from the Higher Education Funding Councils for England and Wales.

This funding was distributed via a series of funding calls. Around 70% of the funding was allocated to addressing the recommendations of the HE Mathematics Curriculum Summit. In order to allow for interesting innovation which could not be predicted, calls for funding always included an open call for projects fitting the National HE STEM Programme aims. Around 30% of the funding allocated was for new innovations discovered this way.

Counting everybody who was named as a project collaborator or as an author in one of the publications (but not those who, for example, spoke at one of the workshops), this represents the work of more than 120 individuals working at 41 UK higher education institutions, two professional bodies, two schools, three non-UK universities and various companies There were over 50 workshops, seminars and conference presentations associated with this work.

The projects we supported have lots to share – good practice advice, evaluated innovative approaches, problem banks and other curriculum resources that you can pick up and use right away, and much more. A Summary booklet provides details of the aims, objectives, outputs and outcomes of each project with Links to access the resources created by each project. Projects are arranged into themed sections.

Developing graduate skills: A booklet was published collecting case studies of successful methods to improve graduate skills development – skills that employers require from graduate employees and academics seek in incoming PhD students – within a mathematical context. Three mini-projects were commissioned based on these case studies and provided evidence that some of this practice is suitable for transfer elsewhere. In addition, mathematics-specific resources and teaching practice on speaking and writing skills were developed and shared.

Engaging with employers: Projects working with employers, employees or professional bodies, either in delivery of a curriculum approach or providing input to develop good practice advice or curriculum resources that you can use. This includes resources giving an idea of what it is like to work as a mathematician and a survey of graduates’ views of the mathematics HE curriculum.

Industrial problems: Banks of real world problems developed in consultation with industrial partners made available for undergraduate projects in mathematics and statistics.

Problem-solving: Two projects working to share good practice and develop curriculum resources on the teaching of problem-solving. We say mathematics develops problem-solving but do we actually know how to develop problem-solving as a skill in our students?

Maths Arcade: An innovative practice involving developing mathematical thinking, providing student support (particularly at the transition to university) and building a staff and student mathematical community. A case study booklet gives details of its implementation at eight universities.

Student-centred Approaches: Projects working to accommodate student needs or taking a student-centred view on improving the undergraduate experience. Including methods for supporting students in different contexts, helping engineers better understand their mathematics and providing adjustments for students with disabilities.

Assessment: A major project conducted research to answer questions about what alternative methods of assessment can offer, evidence of validity and guidance on the process of changing your teaching to involve a new assessment type.

Audio-visual media in teaching and learning: Investigating the recording of lectures and other teaching and learning content, and the effectiveness of learning through audio-visual media.

Projects have completed research and collected good practice advice, developed innovative practice or produced and shared curriculum resources to address various issues in mathematical sciences HE curriculum development. The work includes the need to develop graduate skills and take account of employer requirements, while remembering to ground this in mathematical content and take account of the needs of the discipline. How the collected resources affect the ability of the higher education mathematical sciences community to more effectively develop graduate mathematicians depends on how well these are taken up. This substantial set of projects in curriculum development has produced outputs with the potential to be very useful. Please use them!

The most complete set of links to project resources can be accessed via Mathcentre. The project homepage and this blog also have lists of projects and publications.

You can find out about the wider National HE STEM Programme, of which we were part, by visiting the HE STEM website.


Links to work on HE Curriculum Innovation in Mathematical Sciences

In May I am giving a presentation at four workshops with the National HE STEM Programme Mathematical Sciences Strand. The workshop title is ‘Maths Strand Outputs in the National HE STEM Programme‘ and this will be repeated in Manchester, London, Cardiff and Birmingham. I will be speaking on ‘Work on HE Curriculum Innovation in Mathematical Sciences’. In this talk I will name several resources that my project has produced. These are listed below so I can give an easy link to participants.

Views of ‘young researchers’ on on graduate skills

I ran an exercise at the Young Researchers in Mathematics 2011 Conference in which I asked for participants’ views of the HE curriculum. You can view a video of this session online and I wrote a paper on this: Views of HE curriculum from ‘Young Researchers in Mathematics’ (MSOR Connections, 11(3), pp. 20-21).

Graduate skills development

The booklet Developing Graduate Skills in HE Mathematics Programmes contains case studies of opportunities to develop graduate skills within mathematics curricula.

You can view videos of sessions at the workshop ‘Teaching Students to Write Mathematics’ online and download related materials or burn your own copy of the DVD online.

Engaging employers

The project Assessing student teams developing mathematical models applied to business and industrial mathematics is described in an article in MSOR Connections.

HE Mathematics Curriculum Summit

The Summit findings report is available as HE Mathematics Curriculum Summit.

The papers relating to ‘Student group with industry’ are Student mathematical modelling workshops as preparation for study groups with industry and Mathematical modelling study group.

The report of the sigma-sw summer interns project is available as Summer internships in sigma-sw.

Other innovative work – found through open calls

Find out about the Maths Arcade through The University of Greenwich Maths Arcade.

The inclusive curricula booklet is Good Practice on Inclusive Curricula in the Mathematical Sciences.

Read a report about the project Engineering Students Understanding Mathematics (ESUM).

See videos of talks at the Media-Enhanced Teaching and Learning (METAL) workshops.

Our work until that point was all covered in the HE STEM special issue of MSOR Connections 11(3). This includes final reports from our first call projects, interim reports from our second call projects and initial plans from our third call projects.

Using social media to engage students – a list

I am running a workshop today on using social media to engage students, particularly mathematical sciences undergraduates. I think this is an emerging area about which little is known. I’ve tried to think of some examples of what you might do with these technologies. What do you think of my list? I’d be pleased to hear suggestions for additions, or stories about when you’ve tried this and how it went, in the comments.

Course availability

The University and College Union (UCU) have issued a report Course cuts: How choice has declined in higher education. The UCU press release give the headline figure as:

The number of full-time undergraduate courses on offer at UK universities has fallen by more than a quarter (27%) since 2006… Despite an increase in student numbers.

The press release gives as a key finding that “Single subject STEM courses down 15% and arts and humanities down 14%”.  (Given that, it is a little strange to see Times Higher Education putting the emphasis solely on arts and humanities courses.)

The focus from UCU is on choice, with general secretary Sally Hunt quoted saying:

This report shows that, while government rhetoric is all about students as consumers, the curriculum has actually narrowed significantly.
If we want to compete globally, we simply cannot have areas of the country where students do not have access to a broad range of courses.

This focus on geographical differences puts me in mind of the Steele report, “Keeping HE Maths where it Counts” (2007), which took an interest in the regional availability of courses with a broad range of entry requirements and had a finding about “mathematical deserts”, areas where students tended to stay local for university where mathematics is not available as an option.

Key findings aside, I was struck by the per subject data for course availability. I am not as aware of cross-subject comparisons as I should be. I am used to hearing a complaint that the professional and learned bodies in mathematics (IMA & LMS) only have about 7 thousand members, compared to forty or fifty thousand each for IOP and RSC, despite mathematics graduating nearly as many students per year as physics and chemistry combined (and what this says about how mathematics undergraduates view themselves as part of a wider mathematical community). This gives me the idea that mathematics is a widely available subject compared to others. This, it seems, may be a fallacy.

The table below is a reduced version of this table, which I compiled from data given in the Course cuts report. The totals refer to degree course provision in the UK. I have taken the liberty of combining a few lines from the original report. There were some subdivided disciplines with relatively few courses. I may, in my ignorance, be committing a sin as terrible as combining biology and computer science as the same, but I have combined three courses on history, two on law, three modern languages and two classics into single lines. These combinations are indicated in the table. I hope these are reasonable.

Subject 2012 total decline since 2006 decline as percentage of 2006 total Proportion of G100 Mathematics
I100 Computer Science 169 38 18.36% 2.49
N100 Business studies 151 11 6.79% 2.22
Q300 English studies 116 -4 -3.57% 1.71
Law: M100 Law by area & M200 Law by topic 145 6 3.97% 2.13
History: V100 History by period, V200 History by area & V300 History by topic 143 17 10.63% 2.1
L300 Sociology 92 14 13.21% 1.35
C100 Biology 88 11 11.11% 1.29
L200 Politics 79 2 2.47% 1.16
H200 Civil engineering 73 -2 -2.82% 1.07
L100 Economics 71 9 11.25% 1.04
G100 Mathematics 68 7 9.33% 1
F800 Physical geographical sciences 65 21 24.42% 0.96
F100 Chemistry 59 3 4.84% 0.87
L700 Human & social geography 50 13 20.63% 0.74
F300 Physics 47 -3 -6.82% 0.69
V500 Philosophy 45 3 6.25% 0.66
Modern languages: R100 French studies, R200 German studies & T100 Chinese studies 77 15 16.30% 1.13
Classics: Q600 Latin studies & Q700 Classical Greek studies 16 1 5.88% 0.24

In case you are interested, the numbers for the subjects the report claims the decline is most particularly in are:

For STEM: biology (down 11 to 88 courses, an 11% reduction of 2006 numbers), physical geographical sciences (down 21 to 65, a 24% reduction) and computer science (down 38 to 169, an 18% reduction);
For social sciences: human and social geography (down 13 to 50, a 21% reduction) and sociology (down 14 to 92, a 13% reduction);
For arts and humanities: French studies (down 10 to 26, a 21% reduction), German studies (down 6 to 21, a 17% reduction) and history by topic (down 13 to 34, a 27% reduction).

What I am most struck by is the number of courses still available for some subjects. Having thought mathematics was relatively available, based on a comparison with physics and chemistry,of the subjects included in the report I see only physical geographical sciences, chemistry, human and social geography, physics, philosophy and classics are less available. (I wondered about combining physical and human & social geography, which would take geography above mathematics, but decided against it because the report classified one as STEM and the other as Social sciences.)

There are more degree courses available to study in each of computer science, business, law, history, English, sociology, biology, politics, modern languages, civil engineering and economics than in mathematics.

Perhaps this isn’t unreasonable. Of course, there is always going to be variation and the availability will be demand-led, but when I see that an applicant wanting to study computer science, business, law or history has more than twice as many options as those wishing to study mathematics, and that mathematics is the twelfth most available subject out of eighteen in both 2012 and 2006, I can’t help feeling a little sad for my discipline. (Of course, the picture is even worse for other subjects; those I have listed as twice as available as mathematics have more than three times the number of physics courses available.)

‘Alternatives to lectures’

I have been asked to speak at the third Media Enhanced Teaching and Learning (METAL) Workshop at the University of Nottingham on 11th January 2011 with the title “Alternatives to lectures”. This series of four workshops is part of a project in using media in teaching and learning at the University which my project is supporting. Here, roughly, is what I will say.

At the first METAL workshop I spoke about effectiveness of lecture capture. You can watch a video of this as Further uses of screencasting – but are they effective? or read a write-up as Lecture capture technology – technically possible, but can it be used effectively?

As part of that talk I looked into the link between use of lecture recordings and achievement. One study identified as a positive behaviour as students coming to class then using the video recording to revisit points they struggled with. On the other hand, skipping lectures to watch the videos instead seemed to be a detrimental approach.

I also considered what might be the effect of lecture capture on attendance. The studies I found seemed to indicate a split here. Traditional, non-interactive lectures where the students watched, listened and copied what the lecturer wrote on the board observed a decrease in attendance. Those lectures which included an interactive component did not observe such a decrease in attendance. The implication might be that if the video recording faithfully replicates the lecture experience then students see little point in attending.

These results, taken together, seem to suggest that increasing interactivity in lectures encourages students into the positive behaviour mode. A few things are being conflated here and it’s all based on small scale studies, but a question is raised about whether traditional lectures are really that effective. My talk tomorrow will draw on this theme to suggest methods to increase interactivity.

The direct inspiration for this topic being on the workshop schedule is an American RadioWorks documentary Don’t Lecture Me, part of a series on 21st century ‘college’ (in the American sense) education.

Part of this talks about students’ preconceived ideas about the physical world and the effect this can have on their understanding of physics, saying:

One reason it’s hard for students to learn physics is that they come into class with a very strong set of intuitive beliefs about how the physical world works… It turns out though that many of these intuitive notions do not square with what physicists have discovered about how things actually work. Most people’s intuition tells them if you drop two balls of different weights from the second story of a building, the heavier ball will reach the ground first. But it doesn’t – and this is a very difficult concept for most students to understand because they already have a concept in their mind that’s in conflict with this new concept.

Giving his students a conceptual physics test, Eric Mazur reports:

When they looked at the test that I gave to them, some students asked me, “How should I answer these questions? According to what you taught me, or according to the way I usually think about these things?” That’s when it started to dawn on me that something was really amiss.

This sort of thing isn’t just happening at the applied end of the spectrum; it can happen in pure maths too. I remember reading some work by Lara Alcock and Adrian Simpson, Ideas from Mathematics Education, which discusses students’ preconceived or intuitive ideas of mathematical concepts (“concept images”) – using examples such as functions, limits, groups – and how these are relied on by students above formal definitions, even when the two fail to coincide significantly. Among much else of interest in that book, they say:

Pre-existing concept images might override or interfere with the use of the definition, even when the latter is known.

This brings me to a video I saw a while ago by Derek Muller on the effectiveness of science videos. The part I want to focus on is when Muller studies the responses of students who watch a video passively. In the video, when what is said differs from a participant’s conceptual understanding they don’t notice, their test scores before and after the learning stay the same and they actually become more confident in their misconception.

I’m not sure YouTube has a very thorough peer-review policy and I haven’t read the original research but the idea is interesting. Don’t Lecture Me makes a similar claim about traditional lectures:

The traditional, lecture-based physics course produces little or no change in most students’ fundamental understanding of how the physical world works. Even students who can solve physics problems and pass exams leave the traditional lecture class with many of their incorrect, intuitive notions intact.

There’s a question here about how anyone becomes a physicist. The answer given in the piece is that roughly 10% of students are motivated to teach themselves. David Hestenes is quoted saying: “They essentially learn it on their own”. It may be that the best students (and future researchers) are learning in spite of the teaching, not because of it.

So if simply watching a teacher talk through correct material isn’t helping to challenge students’ misconceptions, what can be done?

Muller advocates presenting students with common misconceptions. In the video he describes an experiment in which participants are shown a video in which their misconception is presented by an actor and then challenged in a discussion with another actor. The participants reported finding the video harder to watch but their test scores increased.

In Don’t Lecture Me (and in life), Mazur advocates a method called peer instruction. In this, students are asked a multiple-choice question in class and allowed to vote on the correct answer via an audience response system. They are then asked to discuss their answer with students sitting near them. If two students’ answers differ then whoever is correct ought to be able to convince the other of this.

What is common about these methods is the use of discussion to challenge misconceptions. Muller uses actors while Mazur uses peers, but in neither case does an authority figure tell anyone the correct answer wholesale. I’d say using discussion to challenge misconceptions is clearly indicated as a potential strategy, with peer instruction the better for a lecture environment.

In Don’t Lecture Me, Mazur says peer discussion works because the peer recently shared the conceptual difficulties. He says:

That’s the irony of becoming an expert in your field. It becomes not easier to teach, it becomes harder to teach because you’re unaware of the conceptual difficulties of a beginning learner.

I expect the approach works because students are evolving their intuitive concept towards the formal version, rather than trying to memorise a second, formal definition in parallel (or in conflict) with their intuitive one. Alcock and Simpson suggest mathematicians are still using concept images to think mathematically, but that they are doing so with “sophisticated images which they can rely on to closely match the [formal] definition”.

A while ago Sally Barton and I did a study of a lecturer’s use of audience response system (electronic voting system, clickers?) questions in class. He took fifteen minutes once a fortnight to present a quiz of five questions to students, with the aim of encouraging students to keep up to date with their lecture notes. After voting on the answers, students were told the correct answer and directed to the module webpage for worked solutions.

First, we asked students to rate on a scale their approach to answering the questions from “I think carefully about the questions asked” to “I don’t think, I just choose answers at random”. The students whose answer suggested they were more engaged with the quizzes reported taking remedial action much more than those who seemed less engaged. However, the ‘more engaged’ students reported that they were able to keep up to date with lecture notes in this module and others (where quizzes weren’t used) equally well. This suggests the quizzes were not needed as an extra incentive to keep up to date for these students. The ‘less engaged’ students tended to take little remedial action, even when they had not known the answer and had simply guessed correctly, suggesting that the quizzes were not encouraging those less engaged students to interact with the teaching materials.

When they would take remedial action, the action taken most often by the ‘less engaged’ students was not to work through the problem again, check the model solution or read lecture notes, but was to discuss the problem with their friends.

We wrote this study up in the proceedings of the CETL-MSOR Conference 2010 as ‘Using an audience response system – what do the audience DO with the feedback?’ (pp. 12-22).

If we’re right, that this group of students are least likely to engage with formal teaching material but perfectly agreeable to discussion with peers, and if this result generalises, then peer instruction could have real positive consequences for these least engaged students.

Visual impairment and inclusive curricula

On Twitter I set up a script to tweet something every day from an archive of tweets. Today it chose to link to a report of a workshop I chaired on Visual impairment in maths, stats and operational research (MSOR). I got a reply thanking me for the link. I won’t say who because the account is private, but this person said this is really useful as they work supporting a blind student. The purpose of this post is to point to a few further links that may be useful.

First, I co-authored Visual impairment in MSOR, a report on a piece of research I was involved with. The report itself may interest and some of the references used may be useful to read.

Next, Accessibility in MSOR: one student’s personal experience may be interesting.

The MSOR Network, my current employer, runs a working group on disability, Accessing MSOR, which operates through a mailing list that you may wish to join. To join requires authorisation, so you should email the group chair Emma Cliffe and introduce yourself so she knows to approve you.

Last year my project supported a workshop on inclusive curricula and Emma Cliffe is currently preparing a booklet based on this workshop ‘Good Practice on Inclusive Curricula in the Mathematical Sciences‘. I will announce on this blog when this is published.

We have also supported a project Methods to produce flexible and accessible learning resources in mathematics. This aims to address an issue arising from the above activity. The description of the project is below:

A curriculum barrier for students with disabilities is the delivery of mathematical learning resources such as lecture notes, problem and solution sheets in inaccessible formats. The current practise of repeatedly re-typesetting notes in to produce particular formats is expensive in the long run. We will develop methods, instructions and examples by which a single master copy may be used to automatically produce a variety of formats. Thus all resources are updated from a master enabling departments to make proactive adjustments. The methods will be appropriate for use by individual lecturers/departments with access to a small range of mathematical/assistive technologies.

Preparedness for mathematics at university

Recently the National HE STEM Programme published ‘Mathematics at the Transition to University: A Multi-Stage Problem?‘, an essay by Michael Grove. This explores research into the preparedness of incoming university students (particularly from A Level) over the last 10+ years and points to some work being done by the Programme at the transition to university to address the underlying issues.

Last year I wrote a blog post over on my personal blog, ‘On the Decline of Mathematical Studies, and ever was it so’, which looked at a few historical reports and wondered whether this was  just a case of each generation thinking the next has declining standards. Having read Michael’s piece and interested that the problem may be being reported in the same way while the underlying cause is shifting, I wrote a follow-up blog post, ‘Shifting decline of mathematical preparedness?’, and recorded a segment for this weeks Pod Delusion.

The Pod Delusion is a weekly radio programme and podcast. On its website, it describes itself like this:

The Pod Delusion is a weekly news magazine radio programme and podcast about interesting things. From politics, to science to culture and philosophy, it’s commentary from a secular, rationalist, skeptical, somewhat lefty-liberal, sort of perspective. A bit like From Our Own Correspondent but with more jokes.

The segment I recorded is about six minutes from 38:30 in Pod Delusion episode 114.

On the subject of work the Programme is doing to support students at the transition from school to university, the Mathematical Sciences Strand (of which my project is a part) is supporting a network of sigma maths and stats support centres in England and Wales. The sigma centres at Coventry and Loughborough have jointly won the Times Higher Education Award for Outstanding Student Support.