matemicaA Graph-Based Hierarchical Model for Mathematical Knowledge Diagnostics
Abstract and conceptual foundation
A primary challenge in mathematical assessment is distinguishing between the point of origin of a cognitive deficiency and its surface-level manifestation. A student's errors in compound percentages or algebraic equations rarely stem from an inability to grasp those specific topics; rather, they arise from unaddressed gaps in foundational prerequisites, such as calculating parts of a number or mastering the order of operations.
This diagnostic framework bypasses superficial symptoms to isolate and identify the underlying structural dysfunction within a student's knowledge base.
A schematic illustration of the diagnostic output, not empirical data. Percentages and the constructs above them were not administered: they depend on a prerequisite already identified as deficient, and their evaluation would carry no diagnostic information.
Key principles of the diagnostic algorithm
Directed Graph of Knowledge Architecture
The curriculum is represented not as a linear sequence but as a strictly hierarchical directed acyclic graph. Each construct is formally bound to the prerequisite constructs upon which it depends.
Adaptive Recursive Testing
The assessment algorithm operates recursively. Upon a correct response, the testing trajectory advances toward higher conceptual complexity. When an error occurs, the trajectory shifts downward toward foundational mechanics.
Filtering of Dependent Concepts
If a gap is detected in a core prerequisite, evaluation of higher-level dependent topics is suspended. These topics are categorized as premature rather than unknown, preventing misdiagnosis and optimizing instructional time.
Differentiated Distractor Analysis
Every incorrect answer option models a specific, named cognitive error — misapplying the order of operations, averaging rates incorrectly. This provides precise data to structure subsequent targeted remediation.
Procedural Fluency and Response Latency
Response speed is an integral metric alongside accuracy. A correct answer delivered with excessive latency indicates that the student is re-deriving the method rather than demonstrating mastery, signalling an unstable foundation below.
Method and scope
The measurement principles applied here — adaptive item selection, distractor analysis, and response latency as an indicator of procedural fluency — are long-established in educational assessment. The instrument itself, including the dependency graph, the item bank and the descent procedure, was constructed for this practice and is calibrated against the responses of its own students.
Its purpose is diagnostic rather than evaluative. The output is not a score and carries no grade: it is a map of where a student's foundation holds and where it gives way, produced so that instruction can begin at the right place.
Built as a working tool for teaching — giving the instructor concrete evidence on which to structure each student's next weeks.