Mathematically Coherent Curriculum

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Geoffrey Saxe
Professor of the Graduate School
Graduate School of Education
University of California, Berkeley


In consultation with mathematicians and mathematics educators, we have developed a curriculum for teaching integers and fractions in the 4th and 5th grades that uses the number line as the principal representational context. A leading idea is that a vector interpretation of the number line is a powerful way to guide teaching and learning of integers and fractions, because a vector represents number as a mathematical object with both direction and magnitude.

Guided by this central idea, we have ordered LMR lessons in a mathematically coherent sequence.

In the first lessons on positive integers, resolution of conflicting ideas lead to definitions for zero is a number on the line, interval, unit interval, and multiunit interval, and every number has a place but need not be shown. In lessons on negative integers, the definition of symmetry on the number line is added, and students investigate how this idea enables them to apply the core ideas of order and unit to negative integers located to the left of 0.

The principles and definitions constructed during the integers unit are foundational for fractions, where, for example, the ideas of unit, zero, and “every number has a place but need not be shown” are key to understanding the definitions for fraction and equivalent fraction.

The fractions unit begins with part-whole and subunit-unit relations as students construct definitions for fraction, denominator, and numerator, and use these to construct and interpret fractions less than and greater than 1. The idea that whole numbers can be expressed as fractions serves as transitional support for lessons involving multiplicative relations. In later lessons, students solve problems that lead to definitions for equivalent fractions and benchmark numbers, and then all principles and definitions are used as resources for ordering and comparing fractions.