# Visual Representations

- Algebra, Geometry

- ISCED 2 = Upper Secondary Education

- English

## Introduction

An evidence-based strategy to help pupils learn abstract math concepts and solve problems is to use visual representations. More than just a picture or a detailed illustration, a visual representation – often called a schematic representation or schematic diagram – is an accurate description of the mathematical quantities and relationships of a given problem.

The purpose of this visual is to reflect the pupil’s understanding of the problem and to help him or her solve it correctly. For example, in the photo on the right, a pupil uses a visual representation – here, a pie chart – to learn about equivalent fractions. Even though teachers implement this strategy in the first grade to help pupils learn basic math facts, pupils with difficulties and difficulties in learning mathematics often do not continue to use it alone to solve problems.

## Learning Outcomes

At the end of the lesson, the learner will be able to:

- Recognize and understand various types of visual representations, such as graphs, charts, diagrams, models, and maps.
- Use visual representations, such as graphs and charts, to interpret and analyze data.
- Identify trends, patterns, and relationships within data sets.
- Make predictions or draw conclusions based on data represented visually.
- Connect visual representations to mathematical or scientific concepts, helping students visualize and internalize those concepts.
- Apply visual representations to solve problems and make informed decisions.

## How Does It Work

Before they can solve problems, however, pupils must first know what type of visual representation to create and use for a given mathematics problem. Some pupils—specifically, high-achieving pupils, gifted pupils—do this automatically, whereas others need to be explicitly taught how. This is especially the case for pupils who struggle with mathematics and those with mathematics learning disabilities. Without explicit, systematic instruction on how to create and use visual representations, these pupils often create visual representations that are disorganized or contain incorrect or partial information.

A move from concrete objects or visual representations to using abstract equations can be difficult for some pupils. One strategy teacher can use to help pupils systematically transition among concrete objects, visual representations, and abstract equations is the Concrete-Representational-Abstract (CRA) framework.

The Concrete-Representational-Abstract (CRA) framework helps pupils gain a conceptual understanding of a mathematical process, rather than just completing the algorithm (e.g., 2 + 4, 2x + y = 27). Systematically connecting concrete objects or visual representations to the abstract equation is a way to scaffold a pupil’s understanding. The components of the framework are:

- Concrete —Pupils interact and manipulate three-dimensional objects, for example algebra tiles or other algebra manipulatives with representations of variables and units.
- Representational — Pupils use two-dimensional drawings to represent problems. These pictures may be presented to them by the teacher, or through the curriculum used in the class, or pupils may draw their own representation of the problem.
- Abstract — Pupils solve problems with numbers, symbols, and words without any concrete or representational assistance.

## Why Is It A Good Practice?

Students learn to reason symbolically, and consequently the complexity and type of equation and problem they can solve increases.

By incorporating visual representations, teachers can accommodate different learning preferences and ensure that all students have access to the content.

Visual representations capture students’ attention and make the learning process more engaging and interactive. They stimulate curiosity, creativity, and critical thinking, motivating students to actively participate and explore the subject matter.

## Assessment

CRA is effective across all age levels and can assist pupils in learning concepts, procedures, and applications. When implementing each component, teachers should use explicit, systematic instruction and continually monitor pupil work to assess their understanding, asking them questions about their thinking and providing clarification as needed. Concrete and representational activities must reflect the actual process of solving the problem so that pupils are able to generalize the process to solve an abstract equation. The illustration below highlights each of these components.

## Inclusion

The exercises can be personalized and at different levels, so all the exercises can fill the differences between children.

Students who use accurate visual representations are six times more likely to correctly solve mathematics problems than are students who do not use them. However, students who use inaccurate visual representations are less likely to correctly solve mathematics problems than those who do not use visual representations at all.

Students with a learning disability (LD) often do not create accurate visual representations or use them strategically to solve problems. Teaching students to systematically use a visual representation to solve word problems has led to substantial improvements in math achievement for students with learning disabilities.

Students who use visual representations to solve word problems are more likely to solve the problems accurately. This was equally true for students who had LD, were low-achieving, or were average-achieving.