What students know (and what they don’t): building a ‘knowledge space’ with fca
We use fca with both positive and negative attributes (e.g., ‘student failed this’) to analyze a math course. This new method finds the ‘key’ items that define the course’s true knowledge structure.
When you design a university course, you have a syllabus. But how do you really know how students learn? Which topic is the one that truly separates those who “get it” from those who struggle? What is the real “knowledge structure” of your class?
In our 2022 paper, published in the International Journal of Computational Intelligence Systems, we used a powerful data analysis method to answer exactly that, using not just what students get right, but also what they get wrong.
🧐 The problem: a map with only half the roads
Traditional data analysis (and basic FCA) often looks at positive data: “student A answered question 5 correctly.” This gives you a map of what students know.
But this is only half the story. Knowing that a student failed question 3 (a negative attribute) is a completely different, and equally important, piece of information. The “map of failure” is just as crucial as the “map of success.”
The problem is, how do you combine both maps (positive and negative) into one single, coherent structure without losing information?
💡 Our solution: “mixed attribute” fca and minimal generators
Our paper introduces a framework using Formal Concept Analysis (FCA) with mixed attributes.
We build a structure that treats “failed question 1” as its own unique piece of data, just as important as “passed question 1”.
Then, we used a concept called minimal generators. In this context, a minimal generator is the smallest set of skills (or failures) that defines a specific group of students. For example, “failing Q2 AND passing Q5” might be a minimal generator for the ‘needs-remedial-help’ group. These are the essential building blocks of knowledge (or lack thereof).
🚀 The results: analyzing a real math course
This wasn’t just theory. We applied this method to the real exam marks from a Mathematics course.
First, we made a key theoretical contribution: we proved the formal, mathematical relationship between the minimal generators of the ‘positive-only’ map and the new, richer ‘mixed’ map.
Second, the analysis of the course data revealed non-trivial insights. We found key items—specific exam questions—that were the real indicators of student understanding. Our method showed how a combination of passing one key item and failing another was a much stronger predictor of a final grade than any single question alone.
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🔬 Why does this matter?
This framework gives educators and e-learning platforms a much more powerful “X-ray” for understanding their courses. Instead of just looking at average scores, they can see the true logical structure of their subject.
It helps answer questions like: “Is it more important to pass Q3 or to not-fail Q7?” This level of granularity is crucial for designing better courses and offering targeted help to students who are struggling with specific key concepts.
📖 The full paper
For all the theoretical details on mixed minimal generators and the full case study on the mathematics course, you can read the original journal article.
Minimal generators from positive and negative attributes: analysing the knowledge space of a mathematics course. Authors: Manuel Ojeda-Hernández, Ángel Mora, Francisco Pérez-Gámez, Domingo López-Rodríguez, Nicolás Madrid. Journal: International Journal of Computational Intelligence Systems (vol. 15, 58)