Hannah got the sweets, who got indigestion?

Last Thursday in the UK around half a million 15 and 16-year olds took a GCSE maths exam, specifically the second paper in the non-calculator exam. By Friday the exam was trending on twitter (#EdexcelMaths), with one particular question attracting attention:

There are n sweets in a bag.
6 of the sweets are orange.
The rest of the sweets are yellow.

Hannah takes at random a sweet from the bag.
She eats the sweet.

Hannah then takes at random another sweet from the bag.
She eats the sweet.

The probability that Hannah eats two orange sweets is 1/3.

(a) Show that n2 – n – 90 = 0

I had a quick attempt and after one unproductive sidetrack I got the answer. So why am I writing about this? Because it fits in with the other posts on assessment I’m doing, and to explore some of the issues around it.

First, the actual  mathematical content is pretty straightforward – you only need to know how to do three things: calculate a probability without replacement, multiply fractions and rearrange an equation. This is hardly Sheldon Cooper territory.

The exam board has two tiers for the qualification (foundation and higher) and probability without replacement is only explicitly mentioned for the higher tier. The exam has been quoted as saying the question was aimed at those students who would achieve the highest grades (A and A*), and I think grade discrimination is a fair approach. I did ask my daughter (who’s currently revising and taking A-level maths) and she said she wasn’t sure she would have been able to answer it at 16. For non-UK readers, GCSE exams are taken at the end of compulsory schooling and A-levels are taken at 18, typically as a route to studying at university.

So why my unease with students finding it difficult? There’s always the charge of dumbing down levelled at exams but I don’t think that’s it. True, when I did my maths exam at that age the syllabus included calculus of polynomials and their applications, which now is only introduced at A-level, but they were different qualifications – GCSEs were only introduced after my school career had ended. I think my unease comes from the fact that I think this shouldn’t have been seen as a difficult question. Donald Clark has blogged seven reasons why he agrees with the children and thinks it wasn’t a fair question, some of which I agree with and most that I don’t.

There’s a couple of factors involved here. I recall reading a study where they looked at who could answer questions with the same maths content but that were written in different ways. That study found that questions written as word questions rather than equations were consistently harder to answer, even though there was no difference in the actual mathematical content. Secondly, I think it’s the way that maths is taught as rules and recipes to follow rather than a creative problem solving activity. This is not a criticism of the teachers because I think that it’s taught that way precisely because of the pressures that have (politically) been placed on education. As I’ve mentioned before I’m a big fan of Jo Boaler’s approach and it’s emphasis on flexibility and application of technique rather than stamp-collecting formulae. Donald Clark makes the distinction between functional maths (maths for a practical purpose such as employment) and the type of maths typically found in exams, but I think that’s a false dichotomy in this case. As Stephen Downes said “… what this question tells me is the difference between learning some mathematics and thinking mathematically.”. The difference between functional and theoretical maths (at this level) starts to disappear when we think mathematically – maths becomes a toolbox of skills to be applied to the problem at hand, rather than a particular formula in a particular topic to be remembered.

And if you’re wondering what the answer was:

The solution to Hannah's sweets

The solution

Times tables – a matter of life and death?

Recently I took my youngest daughter to visit a university in the north-east of the UK, which involved a round trip of nearly 500 miles and an overnight stay. There’s a general election due in less than three months, which means we’re into that ‘interesting’ phase of the electoral cycle where all the parties try to outcompete each other either with bribes incentives for certain groups (‘Unicorns for every five-year old!’) or to outdo themselves with demonising whatever group is the scapegoat this month. If you’ve ever seen the Monty Python’s four Yorkshiremen sketch, you’ll know what I mean.

So what has this to do with times tables? Well, one of the announcements was for every child to know their times tables up to 12 by the time they leave primary school (i.e. by age 11), and by ‘know’ they appear to mean memorise.

I have a number of misgivings about this. Firstly rote learning without understanding isn’t particularly useful. Memorisation isn’t education. Secondly, as the work of Jo Boaler has remarked students perform much better at maths when they learn to interact more flexibly with maths (number sense) rather than than simply remembering the answers. As she points out, calculating when stressed works less well when relying on memory, which is presumably why politicians refuse to answer maths questions when interviewed, as Nicky Morgan the education secretary did recently. In one of my previous jobs I worked in a health sciences department and the statistics on drug errors (such as calculating dosages) were frightening, and there are few things less stressful than someone potentially dying if the answer to a maths problem is wrong.

The outcome of all this memorisation is that the application suffers. As we travelled back there was a radio phone-in quiz and as times-tables were in the news one of the questions was zero times eight. The caller answered eight, and was told they were wrong. A few minutes later someone else called to tell the presenter that they were wrong because zero times eight was eight, but eight times zero was zero. And this is the real problem. While maths is seen (and taught) as a recipe, a set of instructions to follow, misconceptions like this will continue to prosper. Personally, I see maths as more of a Lego set – a creative process where you combine different components in different ways to get to the end result you want. As Jo Boaler has said “When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics”. Unfortunately, I’m doubtful whether that will count for anything against the one-upmanship in the closing months of an election campaign.

Maths and Mindset

An word-based maths problem

An word-based maths problem

Dr Jenny Koenig from the university of Cambridge was the presenter at one of our regular PedR meetings (pedagogical research group) recently. Now, I actually like maths. One of the first Open University courses I did was ‘MS283 An Introduction to Calculus’ so it was interesting to look at maths from a different perspective. The title was ‘Teaching and Learning Maths in the Biosciences’ and dealt with the challenges and issues surrounding quantitative skills in the biosciences, which fell into two main areas. First was content, the mathematical knowledge that a student arrived at university with, which varied according to the subjects and level they studied to and the grades they achieved. What this meant in practice was a very wide range in knowledge and ability from a bare pass at GCSE (the qualifications taken at the end of compulsory education around the age of 16) to a top grade in A-level maths immediately before entry into university. The second area was the attitude to maths, and the issues of maths phobia and maths anxiety. This lead me on to the work of Dr Jo Boaler and her ‘How to Learn Maths‘ MOOC. Unfortunately, by the time I became aware of it the course was due to finish so I downloaded the videos and settled down for some offline viewing. Her book “The Elephant in the Classroom”  is my current hometime reading on the commute home, and goes into the ideas in more detail.
Her premise is that the typical teaching of maths is strongly counterproductive and doesn’t equip students to actually use maths in the way they need to do in real life. This is because it relies on individual work using standardised methods with little creativity or active problem solving. Also, the (predominantly) UK and US practice of grouping students by ability leads to fixed expectations of both student and teacher. Her solution is to use a problem solving approach, involving group work, active discussion and explicit demonstration that there a variety of ways to reach the answer. She draws heavily on the work of Dr Carol Dweck around the concept of mindset, who distinguishes between fixed mindsets and growth mindsets. Fixed mindsets are where a person believes that people possess a fixed amount of a certain trait or talent (like mathematical ability) and that there is little that they can do to change it. This manifests itself as the self-fulfilling prophecy that there are those who are good at maths and those that aren’t. A person with a growth mindset believes that development comes through persistence and practice, and that anyone can improve their skill in a particular area. While these mindsets can apply to any area, I’d argue that Maths is one of the areas where the fixed mindset is particularly common and stated, and not only that, but that it’s culturally acceptable to be bad at maths. For example, while it’s not uncommon to hear people say that they’ve never been able to do maths you’d never see anyone smiling, shrugging their shoulders and saying “Ah, that reading and writing stuff. Never could get the hang of it”. Dweck’s work on mindset really resonates with me, and while I’m largely in the growth mindset there are a few areas where my mindset is more fixed. Now that I’m aware of those I can take steps to change them.
This concept of mindset links in to my earlier post on behaviour and reward because in addition to cultural and institutional barriers to innovation we now can add internal barriers. A fixed mindset leads to risk-averse behaviour because self-worth becomes connected to success. Failure doesn’t present a learning opportunity but passes sentence on the person as the failure. The failure or success at the task is the embodiment of the worth of the individual.
Growth mindsets on the other hand, allow ‘failures’ to be positive. A paper by Everingham et al (2013) describes the introduction of teaching quantitative skills through a new interdisciplinary course, looks at the effectiveness over two years and describes rescuing it “… from the ashes of disaster!” Evaluation at the end of the first year produced some worrying results. Maths anxiety for all students had increased. Female students were less confident in the computing areas of the course and male students were less engaged with the course overall. Significant changes were made to student support and assessment practices for the course and the second evaluation produced much better results. This is a great example of the growth mindset in action – they tried something and it went wrong. Rather than playing the ‘bail out and blame’ game they persisted. They redesigned and tried again, and then made public their initial failure through publication. When I worked as an IT trainer someone asked me how I ran my training room. I replied that I aimed for an atmosphere where people could screw up completely, feel comfortable and relaxed about it, and then get the support to put it right. What works for students works equally well, if permitted :-), for institutions.

References

Etheringham, Y. , Gyuris, E. and Sexton, J. (2013). Using student feedback to improve student attitudes and mathematical confidence in a first year interdisciplinary quantitative course: from the ashes of disaster! International Journal of Mathematical Education in Science and Technology, 44(6), 877–892. DOI: http://dx.doi.org/10.1080/0020739X.2013.810786