Learning Numerical Algorithms

We believe that algorithms and their properties are important mathematical ideas that all students need to understand. An algorithm is a reliable step-by-step procedure for solving problems. To perform arithmetic calculations, children must learn how numerical algorithms work. Some algorithms have been well established through centuries of use; others may be invented by children on their own. The widespread availability of calculators for performing calculations has greatly reduced the level of skill people need to acquire in performing multidigit calculations with paper and pencil. Anyone who needs to perform such calculations routinely today will have a calculator, or even a computer, at hand. But the technology has not made obsolete the need to understand and be able to perform basic written algorithms for addition, subtraction, multiplication, and division of numbers, whether expressed as whole numbers, fractions, or decimals. Beyond providing tools for computation, algorithms can be analyzed and compared, which can help students understand the nature and properties of operations and of place-value notation for numbers. In our view, algorithms, when well understood, can serve as a valuable basis for reasoning about mathematics.

Students acquire proficiency with multidigit numerical algorithms through a progression of experiences that begin with the students modeling various problem situations. They then can learn algorithms that are easily understood because of obvious connections to the quantities involved. Eventually, students can learn and use methods that are more efficient and general, though perhaps less transparent. Proficiency with numerical algorithms is built on understanding and reasoning, as well as frequent opportunity for use.

Two recommendations reflect our view of the role of numerical algorithms in grades pre-K-8:

  • For addition, subtraction, multiplication, and division, all students should understand and be able to carry out an algorithm that is general and reasonably efficient.

  • Students should be able to use adaptive reasoning to analyze and compare algorithms, to grasp their underlying principles, and to choose with discrimination algorithms for use in different contexts.

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