Recent publications in biomaths/stats

  • The Association for Science Education have recently published a really useful guide aimed at secondary school teachers but I think it will also prove to be very useful in HE and FE. It’s called the “Language of Mathematics in Science” and it’s a really useful guide which tries to explain the different language used by maths teachers and science teachers in schools.  Available from





  • And finally…. from last year…  the Association of the British Pharmaceutical Industry published “Bridging the skills gap in the biopharmaceutical industry Maintaining the UK’s leading position in life sciences November 2015” which calls for a greater awareness of the mathematical and statistical requirements in the industry and increased inclusion of maths and stats through school and university.
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Numbas: open source e-assessment system

I have recently rewritten many of the questions from the Essential Maths for Medics and Vets modules using Numbas – an open e-assessment system and this post is about my experiences of using Numbas. I am not a mathematician and not a programmer (though I have done a little programming in the past) – my background is in biological chemistry and I have taught maths for first year biologists, medics, vets and psychology students at the University of Cambridge. The maths questions cover the techniques and concepts that beginning undergraduate students need in physiology, biochemistry, pharmacology and pathology. From what I can tell many of the questions are useful for non-medical biologists too.

What is Numbas?

“Numbas is a web-based e-assessment system that helps users create online tests. It was designed with numerical tests in mind, but is equally suitable for a range of other subjects. … We have created Numbas to be a valuable resource for educators, regardless of their budget. This is why the system is free and open-source, meaning that anyone can edit or adapt it to their needs.

Educators can set up a free account and choose from a public database of questions, or start creating their own. When they have put together their test, they can share it to places where students are already learning. Exams can be shared to a wide variety of locations, online and offline, and can be viewed on tablet, mobile or desktop.[1]


One of the most powerful aspects of Numbas is that you can create variables that change each time a student views the question so they can attempt the question as many times as they like. For example in the following question, the numbers change randomly (within a predetermined range).

(click on the image to enlarge)


And if the student clicks on “Reveal Answer” then can get advice that is tailored to the particular instance of the question:



Randomisation works well sometimes – but not others.

For questions like the one above this works well but for more specific scenarios such as the question below it becomes a lot more difficult.


The tutorials get you up to speed pretty quickly and writing simple questions like these with a few variables was possible after viewing a couple of short videos and reading the documentation.

LaTeX and JME

However…   getting the equations to look good requires the use of LaTeX and this is well known to have a pretty steep learning curve but there is plenty of online help – even a tool that allows you to write a mathematical expression by hand and it displays the LaTeX. Compare the following question display


With the LaTeX


In my opinion setting out the equations clearly is really important, especially for dyslexic or dyspraxic students and particularly for things like fractions and exponents, so LaTeX is really worth learning. The other thing that’s important for this group is font choice. It’s worth noting that any text written within the $ signs that designate the LaTeX input has Times New Roman as its default font. So when you want to include units in with an equation or some variables you need to either put them after the final $ or use the instruction \text to tell LaTeX to display as text rather than as mathematics. I decided in the end that I wanted the numbers to display in Times New Roman so that a one (1) wouldn’t look like an ell (l) but that the prefixes and units would be in the default web font (usually calibri). And whilst on the subject of prefixes, it’s worth knowing that to get a micro sign on a webpage you can go to a website called and type in the LaTeX code for the symbol, in this case \mu and it’ll display the Unicode symbol which you can then copy and paste into Numbas. This is the way of ensuring that the micro displays correctly.

Re-using and copying questions

In my view, one of the most powerful aspects of the Numbas system is the potential to copy, modify and/or re-use someone else’s questions. The first time I did this was when I was trying to figure out how to display standard form. I found a question by Martin Jones (thank you Martin!) and copied it, had a look at how he’d put it together, and then shamelessly created a similar question for myself using the same approach. It’s been quite handy being able to look at other people’s questions – sometimes to just copy and adapt but other times to see how they actually implemented it and to get other ideas for ways of presenting the info or asking the question.

Requiring a student to input their answer in standard form rather than just in scientific notation (i.e. as 1.3 x 106 rather than 13 x 105 or 130 x 104) turned out to be somewhat challenging. Of course 1.3 x 106 is equal to 13 x 105 so the Numbas system considers them the same as it should do. There were occasions when I particularly wanted the answer in standard form.  However with thanks to Christian Perfect, the Numbas developer, the problem was solved. I kicked myself when I saw Christian’s solution as I really should have thought of that myself. If you’re interested in the details, have a look at Christian’s version of my question

Another reason for copying someone else’s question is to add your own context. Because my students are primarily biomedical many of my questions have a biomedical context, so I ask for the concentration of adrenaline solution and express the answer with an appropriate unit or in standard form but it would be very easy to just copy my question and then modify the text to suit whichever field you’re in.

Sharing and Re-using

All of the Essential Maths for Medics and Vets are creative commons licensed (attribution, non-commercial, share-alike – see To download the question sets just go to the following link, select the question set you’re interested in (they have the tick icon in front) and choose which download option works best for you.

This link also contains the original audiovisual tutorials and pdf documents and you can link to this so students can revise the material before and/or after attempting the questions.

Alternatively you can create your own exams by putting together a selection of questions – either ones you’ve created yourself of copies you’ve made of other questions. Questions are generally tagged with keywords so you can find them.


To Christian Perfect and Bill Foster at Newcastle University for their prompt and helpful responses to my queries

To everyone who tested the questions and provided feedback: Nigel Atkins, Kingston University; Rebecca Barnes, University of Sheffield; Chrystalla Ferrier, Westminster University; Dawn Hawkins, Anglia Ruskin University; Robert Jenkins, University of Sheffield; Bernadette Leckenby, Sunderland University; Gemma Marsden, University of Northampton; Rosanne Quinnell, University of Sydney; Lois Rolling, University of East London; Felicity Savage, Anglia Ruskin University.



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Transition Issues

Mathematical transitions: a report on the mathematical and statistical needs of students undertaking undergraduate studies in various disciplines. Jeremy Hodgen, Mary McAlinden, Anthony Tomei. June 2014.  Further information and detailed results of the project are available at the HEA website.

This report covers issues relating to maths in a range of subjects including Business and Management, Chemistry, Economics, Geography, Sociology and Psychology but there is also some interesting data relating to Biology and many of the themes that have emerged are common to all. A few things caught my eye:

1- In 2013 the proportion of students doing biology degrees who had A level maths was 38%, an increase from 26% in 2006 and 35% in 2009 (sourced from UCAS data). In 2013 a further 12% took AS Maths giving a total of 50% of biology students taking some form of post-GCSE maths. It will be interesting to see what effect the decoupling of AS levels from A levels and the introduction of Core Maths will have on these figures.

2- The report recommends the use of diagnostic testing of incoming students. This worries me because, in my experience, if it’s not done well diagnostic testing can be very destructive of confidence and it can reinforce anxiety levels even if it is followed up with targetted support. Research needs to be done on this before it’s implemented.

3- Expectations of both students and academic staff are important and the report highlights the fact that more needs to be done, possibly through outreach.

At the heart of this recommendation, as indeed of this report, is a desire that pre-university students should have a better understanding of what is expected of them and that higher education should have a better understanding of what their new undergraduates can do.

Given that many academics fear reducing student recruitment if they emphasise the maths too much I’m not sure how this is going to come about in practice.


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Biomaths meetings

I think it would be useful to collect together information about meetings to do with biomaths – either research or teaching or both.  If anyone has been to a good conference do let me know (jk111 at – even better, write a short blogpost!

In the meantime there is the BioQuest Summer Workshop coming up, June 21st – 28th  in Delaware. I was lucky enough to go to the 2012 meeting and it was fantastic. The idea is there are a few workshops on specific topics which give you lots of new ideas but the majority of the time you spend working in small teams actually developing teaching approaches or resources for something you want to use in your own teaching. There is some emphasis on maths/data analysis but it is aimed at a typical biology lecturer so it’s about communicating the maths ideas and concepts.

Some of the workshop topics this year include:
1. Interdisciplinary Biology, Mathematics, and Physics Education

2. BIRDD: Beagle Investigation Returns with Darwinian Data

3. Statistical Reasoning in Genomics

4. Two-species Interaction Models

5. Problem Based Learning in Biochemistry

6. Problem Based Learning in Integrated Biology and Chemistry

7. Modeling Phage in a Predator-Prey System

8. Investigative Case Based Learning Online: Big Data on the Chesapeake

9. The Biological ESTEEM Project (Excel Simulations and Tools for Exploratory, Experiential Mathematics): Hidden Markov Model and Evolutionary Bioinformatics

10. NUMB3R5 COUNT: Numerical Undergraduate Mathematical Biology Education

There is more information at the Bioquest website.

The bioquest website has lots of resources from previous years and the ESTEEM and NUMB3R5 COUNT sections are particularly good for biomaths.

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School maths reforms

There’s been a lot going on lately with regard to reform of GCSE and A levels. Keith Proffitt of OCR has written a helpful summary. Keith is a member of the mathematics team at OCR (a UK exam board), responsible mainly for A level mathematics and a new qualification for post-16 students, An Introduction to Quantitative Methods. He can be contacted on 

Key dates

September 2014

  • first teaching of the new National Curriculum in all subjects

September 2015

  • first teaching of new GCSE in Mathematics (and English)
  • first teaching of ‘Core mathematics’ for post-16 students who have succeeded at GCSE but who would otherwise drop maths
  • (first teaching of new A levels in Science subjects)

September 2016

  • first teaching of new A levels in Mathematics and Further Mathematics
  • (first teaching of new GCSEs in science subjects)

A level

A levels are being reformed for first teaching in September 2015 in 13 subjects, including Biology, Chemistry and Physics; the mathematical content in each of these subjects is to be agreed between A level specifications and must be assessed.

Reform to A levels in Mathematics and Further Mathematics has been delayed so they are to be ready for first teaching in September 2016. The reason for the delay is that there are several knotty problems to be solved. The new rules for all A levels say that

  • they must be linear (no module tests, all exams taken at the end of the course),
  • AS level must be a separate exam (so if a student sits an AS level then goes on to sit the A level, all the AS content must be re-examined in the A level examinations).

The problems to be solved include the following.

  • Can A level Mathematics be made linear, and still offer the choice of applications that it currently does (if such a choice is deemed desirable)?
  • Can A level Further Mathematics be made linear, and still offer the choice of applications that it currently does (if such a choice is deemed desirable)?
  • How can AS Further Mathematics be separated from the A levels, when it depends (in some ways) on A level Maths?
  • How can we make these changes without dramatically reducing the participation rates in mathematics and further mathematics?
  • How do we actually address HE concerns about the quality of assessment in A level Maths?

The content of A level Mathematics and Further Mathematics is to be the responsibility of the A level Content Advisory Board (ALCAB ), a company set up by the Russell Group to provide advice to Ofqual on the core content of A levels in facilitating subjects.

It is not yet clear how the other issues are to be resolved.

Core Mathematics

This refers to a working title for the maths courses and qualifications to be designed for all the post-16 students who have passed GCSE maths but who would then otherwise drop the subject. These are planned for first teaching in September 2015. This is an exciting development, a once-in-a-generation opportunity to improve the proportion of post-16 students who do some mathematics, and to make sure that it is the right kind of mathematics. The Advisory Committee on Mathematics Education was asked to advise,, the Awarding Organisations were asked to comment, but the final say about the shape of these courses lies with the ministers at the DfE.

These courses will not be mandatory at first, so drivers to encourage participation will be vital; these could include funding arrangements for schools and colleges, but the attitude of Higher Education will be crucial. Will it become an expectation that students applying for more HE courses will continue with maths post-16?

GCSE Mathematics

New GCSEs in Mathematics (and English) are being developed for first teaching in September 2015. The Mathematics GCSE will be based on a bigger content than currently, and will require more teaching time. The content of the new National Curriculum, for first teaching in September 2014, can be found here

The new GCSEs will have the following attributes

  • Grades will run from 9 (high) to 1 (low). No decision has been made about which new grade is equivalent to a ‘pass’ grade C in the current system.
  • Two tiers for mathematics, with a longer content list for the Higher Tier.
  • Everything is assessed by examination at the end of the course – no modules.

Note that for a few years students will have grades from 9 to 1 in some of their GCSEs – the reformed ones – and A* to G in others.

Currently some schools try to game the system, entering their students for GCSE Mathematics multiple times from Year 10 until they achieve grade C; some schools then allow students who have grade C to give up mathematics, even if they intend to take A level Mathematics later. The DfE has announced steps to tackle these issues.

  • There will be a November examination series for Mathematics (and English) but it will only be available for post-16 students as a resit.
  • The performance of every student at every grade will count towards the school accountability measures, not just those students who get grade C or above.
  • School accountability measures will take into account only the first attempt by a student at a GCSE.

The net effect of these should be that nearly all students sit GCSE at the end of Year 11, and the most able and weakest students get as much attention as those on the D/C borderline.

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A MOOC for How to Learn Math(s)

I can thoroughly recommend the MOOC, by Prof Jo Boaler on the Stanford EdX platform “EDUC115N How to Learn Math.” It is available until Sept 27th and you can start it anytime until then. Whilst I have started other MOOCs and not got past the first section, I found this one so compelling that I stayed up late into the night and pounced immediately when they released the next section.

According to the course info:

The purpose of this course is to help parents, teachers, administrators and others learn important research ideas that will help students learn mathematics effectively.

The course gives an accessible and jargon-free explanation of some of the research in maths learning and gets you to think about how you’d relate that to your own practice, either as a teacher or as a parent helping their child with homework.

Why do I think it’s so important? There are a number of reasons:

1- The course gets you to think about how people get negative messages about maths and why people can be so strongly affected by them. Within this there was an excellent section on stereotype threat which brought in a number of aspects I wasn’t previously aware of. You also consider how powerful mindset is and the sorts of techniques you can use to try to change mindset (this is based on the work of Carol Dweck – ref below).

2. There was a good discussion about how to handle mistakes and some accompanying videos of a classroom teacher putting this theory into practice. This is something I find really difficult – I want to jump in with the right answer rather than guide them towards realising how to get to the answer.

3. There were hints and tips (incl video demonstration) of how to encourage students to communicate their answers in different ways, visual, verbal, diagrammatic, symbolic etc. This is a really important skill which is often neglected. It also showed the importance of getting students to explain different ways of reaching an answer and valuing those different perspectives.

4. The course considered how to help develop a positive attitude about maths. I don’t think I can do it justice here – you really need to watch the videos.

Some excellent references:

Paul Lockhart’s: A Mathematician’s Lament

Steele, Claude M. (2011).  Whistling Vivaldi:  How stereotypes affect us and what we can do (issues of our time). New York, NY:  W. W. Norton and Company.

Dweck, Carol S. (2007).  Mindset:  The new psychology of success.  New York, NY:  Random House.

Jo Boaler’s book “Elephant in the Classroom” covers much of this course’s material.

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Does authenticity help?

Poladian L (2013) Does authenticity help in engaging life sciences students in mathematical models? IJMEST, 44, 865-876)

The latest special issue of the International Journal of Mathematical Education in Science and Technology (IJMEST) has a collection of papers focussing on Quantitative Skills in Science.  This particular paper caught my eye as I had been thinking about this issue of authenticity in teaching maths for biologists. What exactly constitutes authenticity? Is it in the content of the course, i.e. using biological examples to embed the mathematics or is it in the teacher, i.e. someone who uses maths within biological research? Or perhaps it is both?

Poladian tracks the move in traditional mathematics service courses towards situating the maths within realistic and relevant scenarios. But what Poladian points out, and what I think is missing from a lot of maths for biologists courses and textbooks, is the idea of teaching through mathematical modelling, showing that the processes of modelling are of greatest importance and using modelling as the central theme of the course. These processes include developing tools which can interpret and solve a real-life problem and making connections between concepts and procedures. An interesting example of this is described: using the same model in two different situations (spread of an epidemic and population growth in an ecological niche) and two different models for one biological scenario. This allows students to see the mathematics that underpins the biology.

The paper describes ways in which ideas relating to authenticity and realism were developed and implemented in a first year compulsory maths service course for life sciences students at the University of Sydney. It was a follow-on module from an Applications of Calculus course and focussed on models based on differential or difference equations for students intending to major in biology, psychology or medical sciences. Authentic contexts included examples from ecology, pharmacology and epidemiology and did not include many of the usual physics-based examples such as springs, pulleys and motion. Interestingly the mathematics was shown in both context and in context free forms. Student survey responses showed that many students were positive about the choice of contexts and this helped some, though not all, students to see the relevance.

In addition to this choice of authentic contexts there was clearly a contribution at a more emotional level. The enthusiasm of the lecturer and his ability to make connections with current research and items in the newspapers were also important. This suggests that, in order to teach mathematics for biology students, a mathematician needs to show enthusiasm for the biology and the potential for mathematics to enhance biological research.

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Post-16 maths for employability

The Sutton Trust have just published a report: “The Employment Equation: why our young people need more maths for today’s jobs” by Professor Jeremy Hodgen and Dr Rachel Marks of King’s College, London which demonstrates the need for young people to continue to study maths after GCSE. Of particular interest are two of their recommendations:

5. In general, students with at least a grade C at GCSE have already covered the critical mathematical techniques and concepts, but they do need to understand what they already know better. Any specialist mathematical techniques can be learnt in the workplace, provided students understand and can apply GCSE mathematics. The curriculum should also include more “simple maths in complex settings”, by providing students with problem-solving opportunities involving “messy” contexts that do not have straightforward solutions. Students should have many more opportunities to collaborate and discuss, working together to understand, interpret and communicate the mathematics they are involved in.

6. To allow students to more easily transfer their mathematical skills into the workplace they should use computers extensively, particularly spreadsheets and computer-generated graphs, to apply and learn mathematics. Competence in these skills matters in the workplace.

“Simple maths” is defined as:

…[that]  almost wholly within the GCSE curriculum, covering the core areas of:

  • Number, particularly mental maths, approximation, estimation and proportional reasoning
  • Using and interpreting calculators and spreadsheets
  • Statistics and probability, including data collection, interpretation and representation
  • Algebra, particularly graphical representation and diagrams
  •  Geometry and measures, including 2D and 3D representation

The report also notes the increased use of technology, particularly spreadsheets and automated approaches, which has led to a “black-box” mentality where employees have little understanding of what the software is doing. Therefore a combination of developing understanding whilst using software tools extensively in real-world problems  is recommended.

To what extent would this vision of post-16 maths help students entering bioscience degrees? It would certainly be better than the current situation where the majority of bioscience undergraduates  don’t take any maths during their A level studies and thus become rusty on the maths that they had done at GCSE.

However there are some additional  topics that are really essential for bioscientists namely scientific notation, exponentials and logarithms. I think you could argue that basic calculus in the sense of understanding rates of change could also be included. Many mathematics educators argue that bioscientists should at least do AS maths as they would then cover logarithms and basic calculus but then bioscientists argue that it could become difficult to recruit sufficient students and after all it’s really not that important. Or is it?

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More equations = fewer citations: part two

This post was triggered by reading Michael McCarthy’s blog post on “Statistical Inference in Ecology”. In it he asks about writing a chapter on statistical inference for an ecology textbook and is trying to tackle that perennial problem: how much maths can you include and still make it readable and understandable?

In section 2 of the chapter “A short overview of some probability and sampling theory” I thought the issue boiled down to the question:

When writing for a general bioscience readership should you use mathematical symbols such as these?integral

First there is the argument that equations put biologists off because of maths anxiety. There may be some truth in that but I think there is another argument. A sizeable proportion of biologists in the UK have never studied calculus so have no idea what an integral sign means. In the UK I’d estimate this proportion to be about two-thirds based on the maths qualifications bioscience students start their degree with and the mathematical content of a bioscience degree (ref). Obviously this varies widely so these figures are estimates but nevertheless a good starting point.

If a biologist hasn’t studied calculus then including an equation such as this is the equivalent of putting a quote in another language. Just the other day I was reading a novel which had lots of quotes of poems in French. I stopped reading because I wasn’t able to translate the French and the author didn’t translate it into English. And it’s a horrible feeling, leaving me pretty fed up and the book  in the recycling bin. It didn’t motivate me to go and learn French I’m afraid!

How would you translate this equation? A graph and narrative explanation is a integral graphpossibility. You could include the equation but provide a visual and narrative explanation alongside it so that the reader gets the message that the integral sign just means getting the area and the x and x+dx at the bottom and top of the integral sign just show you the two boundary points.

In my experience using multiple modes, for example narrative and visual, works best – does anyone have other ideas?

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