# Representations in Math: Why they Matter and How to Use Them Effectively

## Written by Francis (Skip) FennellMcDaniel College

When considering the teaching and learning of mathematics and regularly planning for such learning opportunities, whether the instruction is face-to-face, hybrid, or online, the decisions related to the role of representations as an integral component of your mathematics lessons are most important. Consider the following statement regarding the importance of representations, as a mathematical process, from the Principles and Standards for School Mathematics (NCTM, 2000).

Representations should be treated as essential elements in supporting students’ understanding of mathematical concepts and relationships; in communicating mathematical approaches, arguments, and understandings to one’s self and to others; in recognizing connections among related mathematical concepts; and in applying mathematics to realistic problem situations through modeling. (page 67)

There are a variety of representations which your students should regularly experience as they engage in the mathematics they are learning. Figure 1 presents a model that presents the types of representations students encounter and their possible connections to each other (Lesh, Post, and Behr 1987).

Figure 1

As students become engaged in doing mathematics, the mathematics they are learning is enhanced through experiences with varied representations. The focus here is to recognize the importance of particular instructional considerations as you plan for and use representations. The choices you make regarding student use of the types of representations must consider the following:

• Access: Will your students have access to the representations you expect them to use? In a face-to-face or hybrid classroom setting you may want to have your students keep particular hands-on manipulative materials (e.g. base ten blocks or pattern blocks) in their desks or located close to where you intend them to be used*. For many mathematics tasks you may want your students to work in pairs or triads, thus sharing use of hands-on manipulative materials or creating a group drawing or written response to a mathematics task. In an online instructional setting your students may construct shared visual representations using at home manipulative kits, online manipulatives or collaborative whiteboard spaces.
• Experience: How comfortable are your students in using particular representations? Have they used number lines in a variety of ways to represent and compare whole numbers and fractions (e.g. open number lines; double number lines; vertical number lines)? Do they understand how base ten blocks can be used to represent whole numbers and also decimal amounts? Do they use pattern blocks to compare and add and subtract fractions? It’s important to recognize that “first time” and early users of manipulative materials (e.g. hands-on use of base ten blocks, Cuisenaire rods, pattern blocks, color tiles), number lines, 10 frames, 100 charts, etc. are often, understandably, challenged by the intent and actual use of such representations. Provide time for your students to become familiar with the characteristics and expectations of the representations they will be using. Such experiences are necessary for students as they learn to use representations appropriately and with comfort and confidence. We adults sometimes forget that representational models may not be seen by our students as we might expect
• Connections: As your students are engaged in doing mathematics they will naturally connect particular representations with others. You may not have considered such connections, in a direct way, as you anticipated student engagement in a lesson. For example, you may actually plan for pairs of students to use color tiles or drawings to represent the perimeters of rectangular regions with an area of 24 square inches, suggesting that the student pairs note the perimeters of each of the rectangles and discuss their solutions. One could certainly suggest that this activity has the potential to engage your students in all of the types of representations depicted in Figure 1 (above), clearly demonstrating that mathematical ideas are enhanced through multiple representations.
• Assessment and Feedback: Since making connections among mathematical representations has the potential to deepen student understanding of concepts and procedures while often serving as a connector or link to the problem solving process, it’s very important for such student engagements to be regularly monitored. The use of classroom-based formative assessment techniques will guide the focus of your use of representations as you, for example, observe how students are engaged in using particular hands-on manipulative materials and then connecting to visual (drawing), symbolic (equation), and verbal (discussion) representations. Ongoing use of formative assessment also provides the opportunity to provide feedback to your students related to their understandings and next steps as well as soliciting student feedback regarding their perceptions of what they understand as well as their own assessment of their instructional needs.

In closing, as you regularly plan mathematics lessons that truly focus on your students doing the mathematics they are learning, consider the types of representations (e.g. manipulative materials, drawings, equations, discussions) you anticipate using as more than critical elements of such plans. Think of such opportunities to engage your students as portals to access and deepen understanding and the related byproducts of confidence in and enjoyment of mathematics.

*hand2mind is a great, well-respected source for manipulative materials and related hands-on activities which involve manipulative materials.

References

Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Lawrence Erlbaum Associates.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Author