How MatheZeit uses artificial intelligence

Since large language models became widely available, it is easy to get the impression that artificial intelligence can improve teaching on its own: a task goes to a language model, an answer comes back, and learning follows. For mathematics education, this approach falls short. Before AI can provide meaningful support, it has to be described what is happening mathematically within the task.

The difference from open chatbots

Freely available chatbots have long been used for homework and practice exercises as well. They answer mathematical questions readily and mostly correctly: enter a task and you get the solution path and the result.

For learning, that is precisely the weakness. The task is completed, but the thinking process it was meant to trigger has not taken place. A general chatbot knows neither the learning objective of the task nor the difference between help that keeps a child thinking and an answer that ends the thinking. It is trained to give helpful answers, not to produce learning.

Mazio is built as the counter-model. The tutor knows the learning objective, gives graduated hints instead of solutions, and leads the child back into working on the task. In the moment, this kind of support feels slower than a finished answer; it is, however, the form of help that learning grows out of.

The didactic model behind every task

In MatheZeit, a task consists of more than a question and a solution. During development, every task is described in subject terms: the learning objective, the intended solution strategies, typical misconceptions and calculation errors, suitable representations, graduated support and appropriate follow-up tasks.

This knowledge comes from mathematics education research and from observing real work on tasks, not from a language model. It gives every task a subject structure that goes beyond the visible task text.

Same answer, different lines of thinking

Two children can give the same wrong answer while having thought about it differently; different wrong answers can point to the same difficulty. An evaluation by right and wrong is therefore not enough. The relevant question is: which mathematical idea lies behind this piece of work?

If a child calculates 42 − 17 and arrives at 35, it has probably subtracted the smaller digit from the larger one in both places, a well-known pattern when crossing the tens boundary. The didactic task model describes such patterns in advance: which lines of thinking are plausible, which misconceptions are known, and which support fits which pattern.

How Mazio works with this

Mazio, the tutor in MatheZeit, therefore does not start with an empty chat window. When a child makes a mistake, a hard-coded subject evaluation first checks whether the input matches a known error pattern. There is a separate description for this for each task type; the same kind of error leads to different help in a cube-building task than in a division. Mazio answers most situations this way, without any language model being involved.

Only when no known pattern fits does the language model come in. It then receives the mathematical frame of the task: the learning objective, the intended solution paths, the misconceptions taken into account, the permitted forms of help and the kind of feedback that is appropriate in this situation. It does not receive any personal data about the child; it sees the task and the work on it. The language model therefore does not have to decide for itself how mathematics is taught, but formulates within a system that has been prepared in subject terms.

The frame for the language model

The flexibility of large language models is their strength and at the same time the reason for this frame. Without specifications, a model could introduce an unsuitable solution path, use a skewed analogy, give away too much or misread a difficulty.

The didactic task model therefore determines which forms of help are permitted, which mathematical terms are used and when a follow-up question makes more sense than an explanation. Two rules apply in every situation: Mazio does not give away the solution, and the last step stays with the child. How the support is graduated is described by the hint ladder in the article on adaptivity.

Two levels, one form of support

Mazio is thus an interplay of two levels. The didactic level describes the mathematical structure of a task along with the known paths and wrong turns. The AI level uses this description to formulate feedback in the specific moment that is understandable and suitable for children.

This interplay makes individual support possible without leaving the mathematical quality to the language model alone. It also keeps the tutor fast: where a prepared response is enough, the child does not have to wait for a language model.

Didactics first, then technology

Many AI applications start with the technology and then look for somewhere to use it. MatheZeit first asks the didactic question of which mathematical thinking processes should be supported, and uses the language model where it improves that support. Mazio therefore starts with the mathematics, not with the AI.