Structured derivations: a unified proof style for teaching mathematics

Abstract

Abstract. Structured derivations were introduced by Back and von Wright as an extension of the calculational proof style originally proposed by E.W. Dijkstra and his colleagues. Structured derivations added nested subderivations and inherited assumptions to this style. This paper introduces further extensions of the structured derivation format, and gives a precise syntax and semantics for the extended proof style. The extensions provide a unification of the three main proof styles used in mathematics today: Hilbert-style forward chaining proofs, Gentzen-style backward chaining proofs and algebraic derivations and calculations (in particular, Dijkstra's calculational proof style). Each of these proof styles can now be directly presented as a structured derivation. Even more importantly, the three proof styles can be freely intermixed in a single structured derivation, allowing different proof styles to be used in different parts of the derivation, each time choosing the proof style that is most suitable for the (sub)problem at hand.We describe here (extended) structured derivations, feature by feature, and illustrate each feature with examples. We show how to model the three main proof styles as structured derivations. We give an exact syntax for structured derivations and define their semantics by showing how a structured derivation can be automatically translated into an equivalent Gentzen-style sequent calculus derivation.Structured derivations have been primarily developed for teaching mathematics at the secondary and tertiary education level. The syntax of structured derivations determines the general structure of the proof, but does not impose any restrictions on how the basic notions of the underlying mathematical domain are treated. Hence, the style can be used for any kind of proofs, calculations, derivations, and general problem solving found in mathematics education at these levels. The precise syntax makes it easy to provide computer support for structured derivations.

Technologies for "Complete, Transparent & Interactive Models of Math" in Education

“…This format is what has been introduced as "structured derivation" [1] long time ago. The format is based on natural deduction [29], the logical foundation of most proof assistants.…”

Which Mathematics for the Information Society?

“…Nevertheless, the novel results mentioned above and preliminary results on the didactical suitability of the calculation format obtained within the group (see [18]) encourage us to continue our efforts. Also, the success claimed by related work like [2] and [26], makes us believe that we can have a positive impact. The possibility of adopting qualitative research procedures, based on case-studies and collaborative action-research projects [22], to assess the proposed methodologies is under consideration if a suitable cooperation with education researchers is achieved.…”

“…High-school students already learn how to solve simultaneous equations on numbers; going from the reals to the simpler boolean domain, where each variable is either true or false, should be no problem. In fact, experiences done in Finland confirm that it is feasible and advantageous to introduce formal logic at secondary-school level [2,4,3]. It is important to note, however, that in these experiments there was no introduction of new mathematics; this means that the use of problems which are outside the curriculum, as the ones shown in sections 3.2 and 3.3, still has to be assessed.…”

Learning how to Prove: From the Coq Proof Assistant to Textbook Style

“…Had we started by trying to prove a conjunction instead we would have received an error-message at the bottom right of the window (see figure 2). 2 In case we make a correct step the actual proof situation is shown at the top right of the window (figure 1). What we know (or assume) can be seen above the line while the goal(s) are listed below.…”

“…Both subproofs can be completed by using our assumptions. 2 This is idealised for the moment. In general suitable error messages have not been implemented yet.…”