## R – Representational

In the representational stage, students use **two-dimensional drawings, diagrams**** or tallies **to solve the same problems as they previously practiced using manipulatives.

Representation may be used for a variety of purposes in mathematical problem solving, such as “summarizing problem information, recording and reasoning about situation/story elements, offloading memory storage, coordinating the results of intermediate calculations, representing numerical or functional relationships via graphs, and making abstract relationships concrete” [i].

Representation is a crucial skill for students of mathematics, as it is one of the mathematical processes identified in the Ontario elementary mathematics curriculum.

Students should be able to go from one representation to another, recognize the connections between representations, and use the different representations appropriately and as needed to solve problems.

For example, a student in the primary grades should know how to represent four groups of two by means of repeated addition, counting by 2’s, or using an array of objects. The array representation can help students begin to understand the commutative property (e.g., 2 x 4 = 4 x 2), a concept that can help them simplify their computations [ii].

However, representation is only a helpful tool for problem solving if it is understood by the learner [iii]. Individuals with LDs can have an especially challenging experience solving problems in math, and research suggests that their use of visual representation strategies differs from their typically-achieving peers in terms of **frequency** of use [iv], **type** of visual representations used [v], and **quality** of visual representations [vi]. However, with explicit instruction, students with LDs are capable of learning representation strategies, such as diagrams [vii].

In this section of the module, you will explore instructional strategies to support students’ effective diagram use for solving problems in mathematics at all grade levels.

The following is an adapted excerpt from the LD@school article* Helping Students with LDs Learn to Diagram Math Problems*. Click here to access the original article.

## Teaching Students to Use Diagrams to Solve Problems

Generating diagrams to solve math problems can help learners in numerous ways [viii]. Early in the process, diagrams can be used as a tool for **recording information** about the problem during the solution process. As the student starts to conceptualize the problem, diagrams can be a tool for exploring alternative ways of **understanding the problem**. Even as the solution is found, diagrams can be used to **monitor and evaluate the solution**.

**Explicit instruction** for diagramming is important for students with LDs because diagrams should be an integrated component of the process of solving the problem, not as a final step [ix]. Without explicit instruction, students may not fully develop their **meta-representational competence**, or the ability to apply knowledge about diagrams to appropriately select, produce, and use diagrams for math problems [x].

To help students to develop their meta-representational competence, teachers should teach the following concepts through explicit instruction:

- What diagrams are
- Why diagrams are used
- When a diagram should be used
- Which type of diagram is appropriate for the math problem
- How to create a diagram
- How to use a diagram

**Two distinct phases** have been proposed to teach students how to use diagrams to solve word problems [xi].

### Phase 1: Understanding Diagrams

Before learning how to use diagrams to solve problems, students need to **learn how to create diagrams**. Furthermore, students need to learn that diagrams are more than simply drawings.

Ineffective diagrams, known in the literature as *pictorial*** diagrams**, depict the visual appearance of variables in the word problem (e.g., a drawing of a character in a word problem). Effective diagrams, known as *schematic*** diagrams**, go beyond visualizing the objects in the problem; they represent the content of the problem and depict the relational information [xii]. Schematic diagrams are extremely helpful for word problems in math and can be transferred across topics in mathematics, including geometry and probability, and across grades [xiii].

### Phase 2: How to Use Diagrams

Once students understand that diagrams are cognitive tools that stand in the place of, and reflect, a thinking process, they need to learn how to use them to solve math problems.

During the second phase, students need to learn that using diagrams requires a **three-step process**:

**Ask**: What needs to be done?**Do**: Produce a diagram.**Check**: Confirm the diagram helps solve the problem.

While these steps may appear linear, students need to know that **these steps are iterative** [xiv].

### Example

The following example of a word problem illustrates the differences between pictorial and schematic diagrams, and **demonstrates how effective diagrams are tools for solving the problem.**

*The comic store is 4.4 kilometres west from Vera’s house. The video game store is 2.8 kilometres west from Vera’s house. How far is the comic store from the video game store?*

*Pictorial Diagram:*

Students with LDs may make the mistake of focusing **too much attention on the details** of the drawings. This student may have spent too much time drawing the buildings and including extraneous information, such as the compass. The unnecessary attention to detail may have been why the student mislabelled the distances.

*Schematic Diagram:*

This student used a diagram that includes measures of length. By including relational information such as the distance on the number line, this student was better able to **conceptualize and solve** the problem.

## References

[i] Zahner & Corter, 2010, p. 180

[ii] Ontario Ministry of Education, 2005, p. 16

[iii] Jitendra, Nelson, Pulles, Kiss, & Houseworth, 2016

[iv] Montague, 1997

[v] van Garderen & Montague, 2003

[vi] van Garderen, Scheuermann, & Jackson, 2012

[vii] Matheson & Hutchinson, 2014

[viii] Stylianou, 2010

[ix] van Garderen, 2007

[x] van Garderen, Scheuermann, & Jackson, 2012

[xi] van Garderen and Scheuermann, 2015

[xii] van Garderen, 2007

[xiii] Zahner & Corter, 2010

[xiv] van Garderen & Scheuermann, 2015