# CHYPRE : an interactive environment for elementary geometry problem solving

Many teachers confuse often reasoning and proving when they teach geometry problem solving. The teacher presents a model of a demonstration (the best of course) and he hopes that students can reproduce it on other cases, perhaps by analogy. He rarely teaches the methods used to find the demonstrations. Students are evaluated on the final writing of a demonstration and we never see anything of their capacities for reasoning.

The use of computers could give more liberty to students to obtain the solution of a problem according to their capacities. Some ITSs (Intelligent Tutorial System) were built in this way. However, when using these products, the student is generally constrained in a "step by step" model of problem-solving. He has to find out each rule necessary to the proof and apply it on existing statements to construct a correct inference. Then the proof is complete when the givens are connected to the goal. Nevertheless, we observe that experts do not go through all the steps of a proof. They focus on the key steps and skip the less important ones. Accordingly, we built "Chypre".

Chypre is an interactive system which helps students with elementary problem solving in geometry courses. It is based on the relevance of basic configurations (e.g. a parallelogram) in a figure. The student designates directly on the figure one configuration. After looking the correctness of the fact, the system completes the logical network and creates "implicit", i.e. facts which can be inferred naturally from the fact designated by the student. For example, an implicit of the fact "ABCD is a parallelogram" is the fact "AB and CD are parallel".

The knowledge organization of Chypre with implicit allows the abstract level used by an expert.

The value of the facts can be:

1. Hypothesis
2. Conjecture
3. Proved (if the fact can be inferred from hypothesis)

The values 1 and 2 are given by the student. The value 3 is evaluated automatically by Chypre. These values are visible for the student who can follow their changes.

The solution representation is obtained by a graph which only contains the relevant elements of this solution. This graph is a reduction of the full resolution graph. It only represents the elements really declared by the user and not the implicit elements created by the system. This graph is in accordance with principles which assure the consistency of the presentation with regard to logic deductions it induces.

The student is absolutely free to explore the problem. He can begin with inferences from hypothesis (top-down) or from goal (bottom-up) or with a fact in the "middle".

We believe that Chypre is approaching an expert model better than a "step-by-step" model. Chypre intends to teach how to reason.

System requirements

• A Personal Computer with a 386DX or higher processor (486 recommended).
• Microsoft Windows 3.1 or Windows 95
• 2Mb of RAM
• a mouse device

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