Using Multiple Representations to Facilitate Conceptual Understanding

So, which type are you? Do you prefer the visual beauty of a graph? Or perhaps you appreciate the particular elegance of an equation. Whether you prefer a graph, symbols, tables, or words, they are all ways to represent a mathematical relationship. The trick here is to give your students opportunities to examine and create each of these representations. Not only does it help them to develop a connected knowledge of mathematical concepts and procedures, it will come into play as they learn about the mathematics of linearity, a concept that they will study in greater depth as they move into algebra and the study of functions.

Check out the following sections of the article, "Key Aspects of Knowing and Learning the Concept of Function," by Marilyn Carlson and Michael Oehrtman:

One of the most critical aspects to learning about functions is for students to understand connections among our previously mentioned modes of representation: symbolic, tabular, graphical, and verbal. This is something that Ms. England knows all too well--her Stacking Cups activity helps students begin to form richer cognitive connections that will support their later learning.

The Stacking Cups task requires that students not only create tables, graphs, and equations to represent their data, but also that they communicate their findings to the class. Let's listen in as students discuss within their group about the presentation in "Multiple Representations".


As students discuss their role in the presentation of their poster, a discussion of the meaning of slope and intercept inadvertently takes place. The students are able to draw connections among the concrete (cup characteristics--base height and lip height), graphical (data on the x-y coordinate plane), and algebraic (equation in slope-intercept form) representations. Particularly noteworthy is how the students continually question one another, pushing for further clarification and understanding.