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
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
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.