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Visual Representations

All students can benefit from using visual representations, although struggling students may require extra support and practice. Visual representations are a powerful way for students to access abstract math ideas. Drawing a situation, graphing lists of data, or placing numbers on a number line all help to make abstract concepts more concrete, whether done online or offline. Developing this strategy early will give students tools and ways of thinking that they can use as they advance in their learning of more abstract concepts.

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Choosing the “right” visual representation often depends on content and context. In some contexts, there are multiple ways to represent the same idea. Your students need to view a variety of visual representations (see UDL Checkpoint 2.5: Illustrate through multiple media) in order to learn when and why they should choose each one. As you plan how to provide differentiated support and practice, consider ways technology tools can support the development of this critical practice.

Steps:

Step 1: Provide Clear Explanations

  1. Check students’ understanding of a visual representation to determine a starting point. Ask them about features, including labels and scales when appropriate. Review any features they are unsure of and encourage them to ask questions.

  2. As students use a variety of online tools to create visual representations (see UDL Checkpoint 5.2: Use multiple tools for construction and composition), ask questions about the format of the representation to ensure that students understand all the features.

    Technology tools for visual representations
    • Spreadsheets to provide structure for tables
    • Graphing calculators or computer-based applications (such as GeoGebra) to quickly create accurate graphs
    • Virtual tools (such as GeoGebra) to create geometric figures
  3. Some representations, such as graphs, use more than one dimension. Highlight for students—or have students tell you—what each dimension represents (see UDL Checkpoint 3.2: Highlight patterns, critical features, big ideas, and relationships). When your students first learn how to use a representation, you may need to do this each time they use it. As students become familiar with the representation, you can check in with them to make sure they can interpret the dimensions.

View evidence behind this recommendation

IES Recommendations

Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

Evidence:
Strong

Source: IES Practice Guide: Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools

Interventions should include instruction on solving word problems that is based on common underlying structures

Evidence:
Strong

Source: IES Practice Guide: Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools

Teach number and operations using a developmental progression.

Evidence:
Moderate

Source: IES Practice Guide: Teaching Math to Young Children

Step 2: Give Students Strategies and Models

  1. When possible, include alternative visual representations and discuss the similarities and differences between the representations. This will help your students to move more easily from one representation to an alternative they prefer. It will also help them recognize limitations in one visual representation that may not appear in others.

  2. Vary the shapes and orientations of representations so that students focus on the important features only as they learn about the objects and situations represented (see UDL Checkpoint 3.4: Maximize transfer and generalization).

    Ways to vary representations
    • Fractions should be represented not only by circle (“pizza” or “pie”) diagrams, but also by rectangles, sets of objects, arrays, lengths (such as fractions of an inch or centimeter on a ruler), and locations on a number line.
    • Geometric objects should not always be shown using regular figures. A quadrilateral should not always look like a rectangle or parallelogram. Triangles should not always have the same orientation (for example an equilateral triangle with a horizontal base always on the bottom).
  3. When you introduce a visual representation for a task, explain the connection between the problem and the representation. Use the same language with the representation that the problem uses.
View evidence behind this recommendation

IES Recommendations

Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.

Evidence:
Moderate

Source: IES Practice Guide: Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools

Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.

Evidence:
Moderate

Source: IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten Through 8th Grade

Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

Evidence:
Strong

Source: IES Practice Guide: Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools

Help students understand why procedures for computations with fractions make sense.

Evidence:
Moderate

Source: IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten Through 8th Grade

Step 3: Provide Ongoing Formative Assessment

  1. Show your students a specific representation—a graph or table—that is missing an important feature (see UDL Checkpoint 3.2: Highlight patterns, critical features, big ideas, and relationships). Ask them to identify the missing feature. For example, they should be able to identify when a graph has no scale indicated on an axis, or that the quantity represented by the axis is missing.

  2. When students get stuck as they try to interpret a visual representation, ask questions that guide their thinking (see UDL Checkpoint 3.3: Guide information processing, visualization, and manipulation). Questions may help you to discover the source of a student’s confusion. Prompt the student to focus on the information the visual representation provides.

  3. Consider each student’s needs and learning styles when you decide which actions to take to move students closer to the learning goals. Give students time to ask you questions, share their thinking, and respond to the feedback you provide.

    Key questions to elicit student thinking
    • Why do you think that?
    • How do you know that is correct?
    • How does that picture represent the problem?
    • Can you explain your answer?
    • Is there another way you could do that?
View evidence behind this recommendation

IES Recommendations

Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review

Evidence:
Strong

Source: IES Practice Guide: Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools

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