Structured Derivations

Structured derivations is a method for presenting mathematical arguments in a precise and easily understandable form. Structured derivations make it easier for students to follow and understand when the teacher presents a mathematical argument. It gives the students a template for how to construct their own solutions to mathematical problems. The uniform presentation format makes it easy to check and find errors in students' solutions. A structured derivation can also be analyzed by a computer,  to check that the derivation is meaningful and that each step in the derivation is mathematically correct.

The format can be used for all kinds of mathematical arguments: calculations, solving equations, simplifying expression, proving theorems, geometrical constructions, and so on. Structured derivations can be used at any level of mathematics, from pre-algebra to university level, and can be used in any area of mathematics.

 

Advantages of Structured Derivations

  • Shows logical structure explicitly
  • Each derivation step is justified
  • Easy to check correctness of derivation
  • Well-defined syntax
  • Supports automatic correctness checking
  • Works for all kinds of mathematics
  • Has a firm logical basis

Example Calculation

This is a standard structured derivation calculation. We write the justification for a calculation step in a line of its own between the two expressions. The justification is written inside curly brackets, next to the equality sign.

The advantages of writing a justification in this way:

  • there is plenty of room for the expression, may take two or more lines,
  • there is also plenty of room for the justification, may also stretch over many lines, and
  • the relationship between the expressions is written out explicitly.

Structured Derivation Video

Solving Word Problems

Example Proof

This is a proof of a well-known trigonometric theorem.

  • The proof starts with the task to be solved (after the bullet).
  • The assumptions that we are allowed to make are listed and labelled (in parentheses).
  • The definitions and mathematical facts following from the assumptions and general theories are labelled (in braces).
  • The ⊩ -sign starts the derivation.
  • The empty square ends the derivation.