Introduction
This is deliberately a very short article.
Maths teachers - teaching the final year of high school - reading this article are urged to try stack, first at a demo site such as
stack.bham.ac.uk
(Our experience suggests that it is best to register as a new user, as the guest access without registering seems limited.)
There is a lot of information about stack at this stack.bham site or sites linked to it, and there is no need to repeat it here. Some repetition is unavoidable, for example, that servers for moodle/stack can be linux or Windows.
In some places near the end of this article the focus is on getting stack in use in WA Schools. If you are from another State, ignore those places. The message that stack is worth looking at remains equally valid.
If, after trialling stack at a demo site, readers with access to a server with moodle - or knowing its site administrator - and who wish to take it further, "IT may be Time" to download and install stack, and trial it on the local server. The cautious "may be" is that the software is continuing to evolve, and the installation instructions for the latest versions are sometimes less detailed than ideal. As at April 2012, there is no longer support for stack 2.0, so use 3.0 or later.
Included in the talk at the conference will be examples using stack. Some of the initial sections of the paper are applicable to any Computer Algebra (CA) underpinned Computer Aided Assessment (CAA) system. After these we discuss why most universities currently use CAA systems other than stack, and why, in spite of this, stack seems eminently appropriate for moodle-using high schools. The final sections suggest possible steps that might be taken if enough teachers think it is worth setting up local servers.
A couple of typical questions for a CA-underpinned CAA system
The CA system underpinning, of course, can generate random numbers.
Thus there are many different instances of questions served out to students.
We will denote items which would appear on a student's screen as integers by a string beginning with a letter and ending with an underscore.
Q1. You are given the following 3 points in the (x,y) plane:
(x1_,y1_), (x2_,y2_), (x3_,y3_)
Enter in the solution box the quadratic function of x which passes through these points.
Q2. Invent a quadratic which has a zero at x=x0_ and a maximum at x=xm_ .
Enter your quadratic, as a function of x, in the solution box:
These, and/or similar questions, will be demonstrated at the talk.
The vast majority of the Year 12 (final year, WACE) Calculus exam questions could be coded into stack. Indeed, a sensible starting point for trialling stack would be to do this and provide the questions as optional revision questions for year 12 students. However, there are whole classes of questions, e.g. the "invent an example" type of question, which are appropriate in systems like stack but would be horrible for teachers to mark (as there are many correct answers, and many ways to get part of the way to an answer -e.g. in Q2 above to have a quadratic passing through x0_ but failing to have its maximum at xm_). And, of course, the CA can provide graphics appropriate to the data given in the student's instance of the question.
Advantages for students and for teachers
The immediacy - and accuracy - of the response is appreciated by students. For teachers, the real advantage is to reduce the amount of repetitious routine marking of hand-written scripts. However, it should be stressed that hand-written homework is not replaced as the students need to be able to write answers clearly. Therefore, it is suggested that a smaller amount of marking of hand-written homework might be set to be done in a more demanding and targetted way. The CAA systems can display complete worked solutions (and are often programmed to do so). The CAA systems can also provide questions in which the student is required to fill in the intermediate steps (and get them correct before proceeding).
The CA underpinnings
The CA underpinnings of CAA allow students to enter (symbolic) answers, have them
parsed by the CA system,
type-checked by the CA system, with the type-testing augmented when necessary by predicates in the CA supplied either with the CA distribution or the CAA, and then
passed through a marking procedure which can recognize any correct form of the answer.
Sometimes care is needed to achieve this recognition.
There is an extensive battery of testing routines available for all sorts of areas of mathematics.
In stack, maxima is rather weakly typed, but predicates are supplied. maxima predicates end in the letter p (just as Mathematica users are familiar with its predicates ending in the letter Q). For the questions on quadratics above, the first part of the type-checking might apply to the student's answer, stored as sans, for example, the maxima predicate polynomialp(sans,[x]). If this predicate retuns False, for example because the student has entered a polynomial in some other variable, y for example, the student would be told "Wrong type" and given information on the required type. Assuming the polynomialp predicate returns True, further testing, e.g. that the polynomial has degree less than or equal to 2, would follow. In any event, making sure that the student's sans is the correct type is an essential first step for ensuring that when sans is supplied to the code testing that sans does indeed answer the question.
Why did the genre start in universities, and why are CAA systems other than stack so prevalent there?
It is valid for maths teachers to query our suggestion that ITs Time to have a look at stack.
It isn't widely used. For that matter no CAA systems are widely used in high schools.
Furthermore, at universities, where CAA systems are used, stack isn't prevalent: there are many others.
There are big differences between high school and universities in connection with Computer Algebra (CA).
Commercial CA systems of which Mathematica and Maple are major examples, have sold well to universities. University mathematics academic staff mainly use of CA for their research needs. These commercial CA packages are also entrenched for use in appropriately advanced maths classes at universities too.
Some of the academic users of the commercial CA systems, faced with increasing amounts of marking and tutorial work, came to see that Computer Aided Assessment (CAA) underpinned by CA, could be useful. One such system, AiM, dates back to the 1980s.
In any event, university maths classes with enrolments around 500 to 1000 students are not uncommon, and CA-using staff wrote CAA programs using the CA of their choice, or at least one where the university had an appropriate licence. Many such systems were written: few so far have spread to very many universities (AiM, mapleta and publishers' systems - WileyPlus, mathxl, etc. are amongst those that have spread a bit: and stack seems likely to amongst those moodle-using universities without an entrenched CAA system already in use).
Individual schools have much smaller enrolments than universities, and very rarely have, or probably never need, significant licences with the commercial CA vendors. Indeed the authors of this article are convinced that CAA would be uneconomic if developed for an individual high school.
Web delivery of programs underpinned by CA has come later: wolframalpha is an example.
Several of the mathematical Computer Aided Assessment systems preceded web delivery: notable amongst these are AiM underpinned by Maple and the University of Western Australia's calmaeth underpinned by Mathematica.
However these were adapted to be web-delivered in the late 1990s: later CAA systems were web delivered from the outset.
A survey of some of the CAA systems used in universities, and why they are used, is given in [KFGS, 2006].
Even at that date it was clear that integration with Managed Learning Environments, MLEs, was going to be useful. Back then, a selling point of the commercial mapleta system was the fact that it could be used in tandem with the Blackboard MLE (and it can now also be used with moodle).
And, although stack has, at present, fewer universities using it than are using mapleta, AiM, various publisher systems, etc., we predict that the uptake of moodle by some universities (and the British Open University in particular) will increase usage of stack at (moodle-using) universities.
All along stack's developer, Chris Sangwin, has been aware of high schools as possible users of the system.
Stack is now at version 3: version 1 preceded integration with moodle; version 2 had stack connecting with moodle via its "opaque" module (open external access to the question module); version 3 has a closer integration with moodle's question module. Version 3 is, at April 2012, in alpha-test distribution. (Repeating what was said before, don't install Version 2 or earlier.)
ITs Time for some moodle using high-school maths teachers to trial stack; and potential use
That teachers first should trial stack - without involving students - is uncontroversial. Assuming a few teachers decide that stack could be educationally useful for them, below are some comments on pressing it into use with students.
There are two aspects of the admin for using moodle-stack. One aspect is getting the register of students onto the moodle server with the stack questions. The second is having a good database of appropriate questions. Both are only going to function successfully with cooperation between intending users. An author of this article (GK) is prepared to be a contact email putting high school teachers interested in stack into contact with each other.
It seems to the authors that, provided there are enough Year 12 Calculus students at a school, setting up local copies of stack at the local moodle site is reasonable. Our guess is that the one moodle site could handle 100 student users of moodle-stack giving reasonable speed, but it may be a lot fewer if the moodle server is heavily used for other subjects. In any event, if the Year 12 Calculus class has at least 15 students, we suggest using the school's moodle server. Having the students using the moodle server from their school would mean that all the class lists were in place. However, many schools have quite small Year 12 Calculus classes, so that it may well be better for such schools to share a moodle-stack server, and provide the relevant class lists (perhaps one server to serve stack to about 20 schools).
Next there is the topic of getting relevant questions. We recommend the first trials - with real student use - of the system could use carefully selected questions from the existing database. It is a larger task to author one's own questions. It will only be sensible to author one's own stack questions for high school use if several schools get together and share experiences and questions. Having some group maintain a database of well-tested, reliable questions is essential. Keeping the question database to be the same at all the schools is also crucial.
In connection with "local needs" perhaps a focus on questions which fit with the forthcoming national curriculum might help. And longer term, the question database would become a national one.
The future?
As regards stack, and indeed the whole genre, the surrounding software environment continues to develop.
Whether it is methods of presenting maths on-screen, including good practice for mobile devices like android tablets or iPads or similar, or details deep in moodle, stack needs sufficient users to get a bit of funding - but not a lot - to keep up to date.
As regards local high school use, how rapidly usage of stack might develop will depend on many factors.
Cooperation between WA stack-users - perhaps a meeting on it through MAWA, perhaps involvement by maths support people in the Curriculum Council and so on - will play a crucial part.
On our way to the future, ITs Time now for some high school maths teachers to learn more about stack.
The cooperation to develop the use of stack should come from those teachers who decide that it would be worthwhile.
References
G. Keady, G. FitzGerald, G. Gamble, C. Sangwin
Computer-aided assessment in mathematical sciences
http://sydney.edu.au/science/uniserve_science/pubs/procs/2006/keady.pdf