According to a report by the University of London’s Institute of Education, the very best 10-year-old English students are as good at maths as their counterparts around the world, but have fallen behind by around two years by the time they reach their GCSEs.
Cue frothy-mouthed calls for more rigour and tougher exams, presumably since you can’t string people up for not being good at maths, even if it is the only language they understand. Cue also a great deal of “it’s all their fault” finger-pointing and insulting generalisations of the “of course, those Asians value study more highly” variety.
A piece on the New York Times Economix blog casts a story from Inside Higher Ed as a piece of applied game theory. Professor Peter Fröhlich of Johns Hopkins University has a grading system in which
each class’s highest grade on the final counts as an A, with all other scores adjusted accordingly. So if a midterm is worth 40 points, and the highest actual score is 36 points, “that person gets 100 percent and everybody else gets a percentage relative to it,” said Fröhlich.
Can you spot the problem? The Economix posts points out that this allows “at least two Bayesian Nash equilibria”:
Equilibrium #1 is that no one takes the test, and equilibrium #2 is that everyone takes the test. Both equilibria depend on what all the students believe their peers will do.
In equilibrium #1, everyone scores the same mark – zero – and the calibrated marking scheme maps this onto 100%. The students, realising this, arranged a complete boycott and were all awarded grade A.
Alas, this was not a game theory class! Prof Fröhlich has since changed his grading scheme.
Dangerous Curves (Inside Higher Ed) has some detail of how the boycott was arranged (had just one taken the test, all would have been forced to follow suit).
Gaming the System (Economix) discusses the related game theory and economics concepts.
via @Tony_Mann on Twitter.
Dr Aiping Xu of Coventry University is asking staff who have experience of mathematics education in the UK and other countries to complete a short questionnaire as part of a survey of international mathematical cultures for the Higher Education Academy. The questionnaire explains the purpose of the study.
A growing number of international students study mathematics at UK universities. Although mathematics itself may be the same the world over, the subject is learnt within a cultural setting and different countries have different mathematical cultures. The purpose of this project is to try to identify key ways in which the mathematical culture of other countries differs from that in the UK, so that both academic staff and students can be made more aware of these differences and so that appropriate induction can be provided.
I am told the questionnaire should take no more than 15 minutes to complete.
Take the survey: Investigation of international mathematical cultures.
The BBC are reporting that plans to change key subjects, including mathematics, from the current GCSE assessment system to a new, tougher ‘English Baccalaureate Certificate’ and to have a single exam board for each subject are “to be abandoned”.
Further information: Planned switch from GCSEs to Baccalaureate in England ‘abandoned’ at BBC News.
via @RosalindMist on Twitter.
I write to share and invite discussion of something I presented at a conference at Nottingham Trent University last week.
I have been thinking a lot about assessment methods and their advantages and limitations for a chapter I am writing for my PhD thesis. For example, I could set a paper test and mark it by hand, as indeed I set one last week and will be marking it when I finish this post, and this allows me to give a personal touch and assess students’ written work but one downside is that I can’t return marks to students very quickly. I could return marks immediately if I used automated assessment, but then setting the assessment would be more difficult and I may be limited in the range of what I could assess. And so on.
I have been trying to classify these advantages and their paired limitations. My thinking is that by viewing different assessment methods as balanced sets of advantages and limitations we can justify different approaches in different circumstances and, particularly for my PhD, explore the advantage/limitation space for any untapped opportunities, which I won’t go into now (but ask me).
Here is my current list of potential advantages that assessment could access. These advantages are each something that I think that some assessment method can offer. My question is: what am I missing? I would be pleased to receive your thoughts on this in the comments.
- Immediate feedback. This is linked to learning from mistakes, confidence and motivation. It can also prioritise procedural learning over conceptual understanding.
- Detailed, personalised feedback. Though there is much disagreement in what I have read whether a human, who can respond to individual student work, or a computer, which will tirelessly generate worked examples using the context of the question asked, will in practice provide this.
- Individualised assessment. This is achieved through randomisation of questions and is linked to repeated practice, deterring plagiarism, allowing students to discuss the method of a piece of work without the risk of copying or collusion.
- Assessing across the whole syllabus. For example, computers can’t mark every topic.
- Testing application of technique. Whether students can apply some procedure.
- Assessing deep or conceptual learning. For example, open-ended or project work may require a detailed manual review to mark. This is linked to graduate skills development, etc.
- Easy to write new questions. Assume it is easy for a lecturer to write questions that students can answer (it isn’t, but we’re talking principle here). Difficulty is introduced by having to second guess an automated system, or having to second guess students to program misconceptions.
- Quick to set assessments. Assume that writing a test manually takes time. By quickly, I really mean choosing items from a question bank.
- Quick to mark assessments. Assume that marking by hand is not quick, perhaps unless the assessment is very short and student answer format very prescribed, in which case the assessment is limited. This is perhaps linked to problems of consistency and fairness when using multiple markers.
- Easy to monitor students. Clearly marking individual work from every student by hand will give great insight, but here I refer to the ability to gain a snapshot of how individuals and the cohort are doing as a whole with a concept, perhaps very soon after a lecture that introduced that concept has taken place.
- Perception of anonymity. I’ve read that some students are happier to make their mistakes if only a computer knows. This can reduce stress.
- Testing mathematical writing. Clearly requires hand-written work.
- Testing computer skills. Clearly requires use of a computer.
For example, then it might be possible to offer ‘Easy to write new questions’, ‘Assessing deep or conceptual learning’ and ‘Testing mathematical writing’ through a traditional paper-based, hand-marked assessment, but this would preclude, for example, ‘Immediate feedback’.
Similarly, a multiple-choice question bank might offer ‘Quick to set assessments’ and ‘Quick to mark assessments’ at the expense of ‘Assessing across the whole syllabus’ and ‘Assessing deep or conceptual learning’.
And so on. I have loads of these for different assessment types.
My question really is, is there anything missing from my list that might be delivered by an assessment method?
If you were interested in Keith Devlin’s Massively Open Online Course (MOOC) Introduction to Mathematical Thinking in the autumn but heard about it too late, didn’t have time to take part, or signed up but couldn’t keep on top of the course, you may be pleased to hear that Keith has announced plans for the course to run again from 4th March 2013 for 10 weeks. This is longer than the seven week course which started in September.
It’s shaping up to be a busy month for education reform in England. Here’s some news in brief.