# Communicating Mathematical Ideas Using a Variety of Representations

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Many mathematical ideas can be communicated or illustrated using a variety of representations. In this lesson, we'll explore ways you can communicate ideas to your students using written, verbal and symbolic forms as well as visual aids and technology.

## Mathematical Communication

Mathematical ideas can be communicated in a variety of ways using many different representations. For example, you can use verbal , written , symbolic and/or visual representation to convey or expound on a mathematical concept. Technology can also be used to teach concepts.

Using a variety of representations can improve and maintain student attention and motivation and help to foster students' analytical skills and overall higher-order thinking. In this lesson, we'll take a closer look at the ways in which you can communicate mathematical ideas to your students.

## Translating Between Verbal, Written or Symbolic Form

Students need to learn the ability to translate between verbal, written and symbolic modes of thought with fluidity. For example, if a teacher presents a student with a mathematical concept in written or verbal form, the student should be able to write a symbolic representation of the concept. Below, are a couple of examples: one involving a circle's circumference and another involving the speed of a car.

### Circle Circumference

Imagine that you have a circle. You might introduce the concept of circumference verbally by simply saying 'circumference'. We then ask a student to express it in written form. The student might provide a written definition for this concept like: 'the distance around one revolution of the circle.' If you wanted to express the concept in symbolic form, you might display the following on the blackboard: C=2πr, where C = circumference of the circle, π=3.14 (a constant, irrational number) and r = radius.

### Speed of Car

Similarly, we might express the concept of the speed of a car verbally, simply as 'speed' or 'velocity (where velocity = distance / time)' . In written form, we might ask that speed be expressed as 'the distance at which a car travels through a given amount of time.' This may in turn be symbolically expressed as v = Δd / Δt, where Δd represents a change in distance and Δt represents a change in time.

## Using Visual Aides or Representations in Communicating Mathematical Ideas

Visual aides are very important in driving home math concepts, as many students are visual learners. Visual aids might include graphs, tables, media, and art appealing to the tactile nature.

### Graphs and Tables

Graphs can serve as visual representations of a mathematical concept. For instance, we might use a graph of a parabola to model the trajectory of some object through space or we might use a power function to represent the balance in a bank account through time under a compounding interest rate.

### Media and Art

Media and art can help students understand math concepts in application, as it makes concepts interesting, tangible, and visible. For instance, we could use any number of M.C. Escher's tessellations to illustrate the concept of fractals or we could use the Mona Lisa or a picture of the Pantheon to discuss the golden ratio, or Fibonacci sequence.

## Using Technology to Communicate Mathematical Ideas

Technology can be used to strengthen ones understanding of mathematical concepts overall, as technology can be used to make concepts interactive and intrinsically interesting. Some examples include the use of a scientific calculator, Microsoft Excel, or any number of math-based software programs, such as Mathcad.

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