July 10, 2006

More thoughts from 14th March

Follow-up to Looking back to 14th March … from Computer-aided assessment for sciences

In the previous blog we decribed some of the features of Maple TA and WeBWorK presented at the March workshop. Two other CAA software architects introduced their brainchildren at the meeting: Chris Sangwin told us about his System for Teaching and Assessment using a Computer algebra Kernel (STACK) , and Martin Greenhow gave us a roller-coaster ride through his Mathletics program.

STACK
This open source software is designed for intelligent assessment of deeper mathematical knowledge in the growing number of subject areas that require it. Although STACK can deliver standard online question types (e.g. MCQs), its real strength is to handle student–provided answers to questions like these:

1. Factorise the following polynomial into a linear and a quadratic factor and hence find its roots:

x^3-3x^2+3x-2

(where a different equation is generated each time the question is called)

2. Write down a continuous function passing through the points (1,0) and (0,1) with exactly three turning points: a maximum, a minimum and a point of inflexion.

This kind of functionality is made possible by calling a computer algebra system (CAS); currently STACK uses the open–source Maxima system (try it out ). It can not only manipulate students' answers and give responsive feedback but can also help to generate problems randomly from a single template and provide corresponding worked solutions. In his talk Chris gave us some fascinating insights into the challenges that mathematics presents in this area, in particular, how to handle the subtleties of notation in

  • students' submitted answers (fx might mean f times x or the functional value f(x))
  • the CAS (interpreting various positions of minus signs for instance)

STACK tolerates informal entries in student answers (for instance, accepting 3xy instead of 3*x*y) and encourages students to "validate" their answers, in other words, to confirm that the program has correctly interpreted their entry when it displays the formal version.

We are currently exploring the possibility of using STACK for some low–level assessment taken by large numbers of first–year mathematics and statistics students. Its PHP architecture marries well with the Department's learning resources website Mathstuff.

Mathletics
Martin Greenhow and his team of research students have been developing this CAA resource and using it for their teaching and assessment at Brunel University for some years now. Its strengths include

  • well–developed and thoughtful pedagogy
  • large banks of mathematics questions aimed at the A–Level/starting–university zone.

The questions (perhaps better thought of as "question templates") are written in a combination of HTML, MathML, Javascript, SVG script, and can call on a library of functions for such things as displaying a polynomial intelligibly in the conventional way. Responsive feedback is a central feature of Mathletics: question templates include randomised parameters and context-aware alternatives; the feedback of hints and model solutions respects the choice of parameters and context, and responds to student mistakes by using their wrong answers to guess at "mal-rules" or common student misconceptions. Experience at Brunel has shown that this feedback plays a central role in student learning. Although the stand-alone questions can be interpreted in a suitable web browser endowed with the appropriate plug-ins, they are really designed for use with Question Mark Perception, which can deliver sequences of questions, record and analyse students' answers, provide the feedback and so on.

Coding individual questions is a skilled and time–consuming activity, but when the parameters are varied and the context is changed, each template generates thousands of different questions (as many as 1020 for some templates!). Thus the Mathletics framework can create essentially unlimited numbers of different exams on the same set of topics, allowing students to learn by extended practice. Mathletics is responsive to issues of accessibility, gender, and ethnic background.

The demanding requirements of authoring have to be set against the huge searchable repository of existing questions: new–style Mathletics (with randomisation) has about 1500 question styles (each realising to thousands/millions of questions) spanning around 120 topics at GCSE/AS and A–level/university levels 1 and 2. They range broadly across algebra, geometry, calculus (incl. Laplace transforms, differential equations and vector calculus), logic, decision maths, numerical methods, economics applications, probability and statistics . New questions are constantly being developed (as part of the Mathematics for Economics: Enhancing Teaching and Learning Project (METAL) for instance), and Martin is keen to encourage others to join in this creative process.

We are planning to try out Mathletics in the Autumn on a small subset of the first-year engineering students without A-Level Maths. If this pilot is successful, we would hope to use it to support the mathematical needs of the whole cohort later on. We have a site licence for QM Perception and are well placed for this. Although my attempts a few months ago to get Mathletics running on the University network were abortive, one of our postgraduate CAA team members, John Hattersley, is now on the case. I hope to report soon of his success, at least on the well-tried version 3.4 of Perception, which will run another year here; another challenge will be to run it on version 4.2, to which we are upgrading next month. Stay tuned.


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