Mathematical knowledge is conserved in the form of mathematical statements about mathematical objects. It ranges from elementary facts like \(5<8\) and classical algebraic and geometrical results we learn in school to all the contents of university mathematics and current research.
As a more advanced example, we consider (a shortened version of) Banach’s fixed-point throrem:
Let \((X, d)\) be a non-empty complete metric space with a contraction mapping \(f: X \to X\). Then \(f\) admits a unique fixed-point.
Students encounter this result in their first year analysis course. In order to understand its meaning they need to know the concepts metric space, completeness, contraction mapping and fixed-point where each of these notions is based again on other notions. Looking at the details, we find a certain hierarchy: completeness and contraction mapping refer to metric spaces in their definitions while the more elementary fixed-point concept can be formulated independently. Finally, the notion of metric spaces uses various aspects from the theory of real numbers so that it is only usefully developed when certain results about real numbers are available.
Such dependencies are a direct consequence of the recursive rules of mathematics which ensure that new objects and structures are always constructed from existing ones and that new facts are consequences of older facts that have been developed previously.
To support a useful storage of mathematical knowledge, the \(\small \mathbb M\)ATh interpreter maintains a tree structure whose node are called contexts. Each context stores a particular subset of the total knowledge by managing a list of facts and a list of references to objects or structures which are addressed in these facts.
Defining and proving are two basic mathematical actions which enlarge these lists. To be meaningful, they require a specification of the context in focus.
In fact, within such a structured environment, moving the focus to other contexts or adding nodes to the tree appear as further basic actions of knowledge management.
It also turns out, that the idea of temporary contexts is a nice model to describe the mechanism of case distinctions or direct and indirect proofs. More specifically, a proof step starting with certain assumptions can be viewed as a focus shift to a freshly added child context whose lists are filled in accordance with the assumptions.
The subsequent steps are carried out in this context until a “qed” request signals that the desired proof goal has appeared in the list of facts. Then, the focus is shifted back
to the parent node while the temporary child context is removed from the tree.
Generally allowing the initialization of contexts with references to elements that are implicitly characterized by inital (unproven) entries in the fact list has far reaching consequences. These so called settings allow the formulation of structures, theories and mathematical models as essential parts of the total tree of knowledge.
Formulated in terms of \(\small\mathbb M\)ATh, the goal of math education is twofold. (1) It should teach the process of creating trees of knowledge and (2) it should convey particular trees of knowledge with results that have shown to be useful in many areas.
By looking at current practice it appears that goal (1) is taught rather implicitly while goal (2) is pursued with much higher priority.
In order to raise the awareness of goal (1) in teaching and to move the teaching concept from implicit and intuitive to explicit and rational, it is necenary to dissect the total process of knowledge generation and conservation into suitable subprocesses. In other words, the process needs to be modeled with names attributed to all the necessary actions and concepts to achieve the goal. This allows to talk (and think) about the process and eventually to teach it by focusing on and training the separate aspects as well as their combination.
This is the goal pursued by the \(\small\mathbb M\)ATh project.
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