Multiple representations (mathematics education)

Multiple representations are ways to symbolize, to describe and to refer to the same mathematical entity. They are used to understand, to develop, and to communicate different mathematical features of the same object or operation, as well as connections between different properties. Multiple representations include graphs and diagrams, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulatives, pictures, and sounds. Representations are thinking tools for doing mathematics.

Use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities. Estimation, another complex task, can strongly benefit from multiple representations

Curricula that support starting from conceptual understanding, then developing procedural fluency, for example, AIMS Foundation Activities, frequently use multiple representations.

Supporting student use of multiple representations may lead to more open-ended problems, or at least accepting multiple methods of solutions and forms of answers. Project-based learning units, such as WebQuests, typically call for several representations.

Some representations, such as pictures, videos and manipulatives, can motivate because of their richness, possibilities of play, technologies involved, or connections with interesting areas of life. Tasks that involve multiple representations can sustain intrinsic motivation in mathematics by supporting higher-order thinking and problem solving.

Multiple representations may also remove some of the gender biases that exist in math classrooms. Explaining probability solely and only through baseball statistics may potentially alienate students who have no interest in sports. When showing a tie to real-life applications, teachers should choose representations that are varied and of interest to all genders and cultures.

Tasks that involve construction, use, and interpretation of multiple representations can lend themselves to rubric assessment and to other assessment types suitable for open-ended activities. For example, tapping into visualization for math problem solving manifests multiple representations. These multiple representations arise when each student uses their knowledge base, and experience to create a visualization of the problem domain on the way toward a solution. Since visualization can be categorized into two main areas, schematic or pictorial, most students will provide on or the other or sometimes both methods to represent the problem domain.

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