Langrechnen

Langrechnen, literally “long arithmetic,” is a term used in German-language mathematics education to describe extended paper-and-pencil calculation procedures. It covers step-by-step operations on multi-digit numbers, including long addition and subtraction, long multiplication and division, as well as procedures for decimals, fractions, and mixed numbers. The emphasis is on accurate procedural work and explicit use of place value, carrying, and borrowing.

It aims to develop procedural fluency and reliability in arithmetic, building a bridge between informal estimation and formal numerical reasoning. Langrechnen is often taught early in schooling, sometimes alongside or prior to the use of calculators, to help students understand why the steps are performed and how digit positions determine value.

Historically the term appears in 20th-century German educational literature, but the underlying practice predates it. In modern curricula, Langrechnen is sometimes integrated with mental math and estimation, or with digital tools that allow students to check long procedures.

Critics argue that heavy focus on rote procedures can undermine number sense if not supported by conceptual understanding. Proponents contend that orderly long procedures provide a solid foundation for higher mathematics and for situations where calculator use is restricted.

Related concepts include long division, manual calculation, mental arithmetic, and place-value understanding.