STACK is a free online assessment package for mathematics using a computer algebra kernel. It provides a new Moodle question type which is designed for evaluating both numeric and algebraic expressions. Hence the questions can be more sophisticated and can mimic mathematical thinking better than classical closed format questions. The grading of multiple student input fields can be combined to allow for nuanced and individual feedback in so-called potential response trees (PRTs).
STACK also implements well designed randomization components, which can either be used as a means against cheating or to enable students to repeat critical computational steps multiple times with different expressions. Furthermore we can study wrong answers and thereby learn about the way students think about our posed problems, making it easier to come up with useful feedback and hints. This way the systems improves over time and becomes a more and more intelligent tool for student centered learning at an individual pace.
STACK questions are computational by design, and it’s not just mathematics which can profit from that: every field where fluency in hands-on computation is important can use STACK to achieve this goal, e.g. physics, chemistry, engineering, etc. We would like to invite members from other departments to have a look at how their students could have a more successful learning experience.
At D-MATH we have used STACK in two settings so far: we used it to build a database of questions for students to practice independently in open formative quizzes. Secondly STACK is used for weekly learning tasks. In these weekly quizzes students are asked questions with a randomized element. They are graded automatically and contribute to the final grade. In the future we envision STACK also as a powerful tool for online or on-site computer exams.
Here are some examples to make the above a bit more concrete :
Single computations:
A question with one single answer field, usually with some randomized parameter (or randomized function, as above). Suitable to replace short and simple multiple choice questions without giving away the answer in an item.
Converting existing question pools:
It is often possible to turn easy open practice questions into STACK questions. From our experience, as the complexity and difficulty of a problem grow, so does the effort to convert it into STACK. However, it is particularly useful for questions where scaffolding would help the students a lot. A scaffolded problem reduces the cognitive load by asking for the results of intermediate steps and thereby guiding through the maze to find the solution. STACK is particularly suitable for scaffolding through its Partial Response Tree (PRT) mechanism: It can check if a wrong intermediate result was used correctly in the following answers and give detailed feedback as well as partial credit that way. A scaffolded STACK problem is thus a rather lengthy question with several intermediate answers as well as explanations given in the problem text.
The first part of a previously traditional exercise sheet problem...
...is now a slightly scaffolded STACK problem with randomized parameters. In this particular question, a wrong answer in computing z1 or z2 also changes the result for z. By using a Partial Response Tree (PRT) STACK will let the student know if the computation for z = z1/z2 was done correctly even if they did a mistake before, similarly to how a humen grader would give partial credit.
Parametrized MC Question:
Using the computer algebra kernel, it is possible to ask multiple choice questions in which randomized parameters can be used in both the question text and the answer items. This is not possible in regular multiple choice frameworks. Such questions are particularly useful in settings where students could try to cheat by communicating with each other. There is a danger in using randomization: Specific parameter combinations can change the difficulty of a problem significantly. STACK can pre-generate the parametrized answers, enabling the lecturer to check if all parametrizations work well.
Exponents, signs, and somethimes even functions can be randomized in answer items.
Exponents, signs, and somethimes even functions can be randomized in answer items.
To make sure the parameters don't accidentally make the problem easier or harder, STACK will generate a set of 20-30 variants so that the lecturer can check and approve them.
Additionally, we started building what we call training modules:
A collection of problems where students can choose to train a certain method and how much guidance they want. Usually they consist of variations of scaffolded problems as described before.
A training module for computing integrals. Students can choose if they wand to get hints to reduce the cognitive load, e.g. on what substitution function they should use. They know they're ready once they can solve the quiz with all types of problems without hints.
How has «remote» teaching been changed or optimised since the beginning of the pandemic?
STACK questions allowed us to produce online material for students to practice with the possibility of giving individual feedback.
In particular in times of the pandemic it is hard to keep the students motivated and show them that we care. Individual feedback may alleviate this problem a bit. It also taught us lecturers what the students problems were without having the students in our lectures, where one has developed a sixth sense for what is considered difficult or not understood.
STACK enabled us to give better help and support during distance learning. STACK also allowed to keep the learning tasks for a grade bonus. As the quizzes were randomized, the problem of cheating was significantly reduced (both in exercise classes on campus and from home).
What are your thoughts and ideas for the time after the pandemic?
STACK practice material should be offered online and can be used on demand. It allows the students to pace their own learning. We think the STACK quizzes should also be kept, but can very well be done in the exercise class and take away the burden of correction from the TAs and simultaneously decrease the risk of cheating.
Course Description
- Name:
- Analysis I / II
- Description:
- Mathematical tools for the engineer
- Objective:
- Mathematics as a tool to solve engineering problems. Mathematical formulation of technical and scientific problems. Basic mathematical knowledge for engineers.
- VVZ:
- 401-0242-00L, 401-0261-00L, 401-0262-00L
- Department:
- D-BAUG/D-MAVT/D-MATL
- Level:
- BSc, fist year
- Size:
- 1000 / year
- Teaching Power:
- 2 experts
