## STACK Documentation

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STACK is a designed to enable students to answer questions with a mathematical expression, such as a polynomial. STACK enables teachers to design sophisticated computer-aided assessments in Mathematics and related disciplines which give specific formative feedback based on objective properties of students' answers. This moves assessments well beyond multiple choice and other types.

Sensible students do not solve multiple choice problems directly but merely check each suggested answer. This is a serious threat to validity: it is impossible to set many questions as multiple choice items without giving the game away.

Really, the student should provide their answer in the form of a mathematical expression and the system should evaluate its properties. Note the student's answer and the feedback which has been automatically generated below.

More about what we are trying to achieve can be found under the philosophy of STACK.

A demonstration server is available in Edinburgh: https://stack.maths.ed.ac.uk/demo

Note, we cannot use a string match because mathematical questions usually have a large variety of correct answers. Sometimes the correct answer can be given in many different forms. The solution lies in using a Computer Algebra System to power the assessment system. The built-in knowledge of mathematics that the computer algebra system provides opens up entirely new possibilities to computer-aided assessment, of which we will now present a few examples:

Primarily, STACK provides a question type for the Moodle quiz. STACK has also been ported to the ILIAS learning environment. For details of that integration see here. STACK can be integrated into other systems using LTI.

## Equivalent answers of the right form

STACK can mark questions where the correct answer can be expressed in many different forms. In mathematics this is the rule rather than the exception because of algebraic equivalence between expressions, for example $$(x+1)^2 = x^2+2x+1$$. STACK can identify these equivalences. STACK can also establish the form of an answer, independent of equivalence to the teacher's. These two properties are independent.

The system can mark questions that ask the student to provide an example. Here is a simple example of such a question:

Give an example of a function f(x) with a minimum at x=0 and a maximum at x=2.


There are many such functions. Rather than comparing the student's answer to the teacher's answer STACK checks that the answer has the required properties. Giving examples is a higher-order skill that was impossible to assess with conventional CAA systems.

## Intelligent randomisation

STACK can randomise problems in such a way that the level of difficulty is kept constant. For example, if a question asks the student to 'diagonalise' a 2 by 2 matrix, then the system can randomise this problem in a way that guarantees that the answer always contains only integers. The trick is to reverse-engineer the randomised question from a randomised answer. Computer algebra is invaluable to support this process.

## Give feedback and partial credit

For example, in a question like: Give an example of a cubic polynomial with the following properties:

• $$p(0)=1$$,
• $$p(x)=0$$ at $$x=2$$ and at $$x=3$$.

STACK can check each condition separately on the student's answer and assign partial credit accordingly. If the student gave the answer $$p(x) = x^2-5x+6$$ for example, then STACK could reply: Your answer does have zeros at the required points but its value at zero is not equal to 1. You received 2 out of 3 points. Please try again. There are a variety of ways in which feedback can be given.

STACK is a direct development of the AIM system, using the open source computer algebra system Maxima.

## Further information

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