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[Music]

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I am Asha Jitendra, professor of special education

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at the University of Minnesota.

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I have served on the panel

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for Improving Mathematical Problem Solving

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Practice Guide.

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Visual representations play a prominent role

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in problem solving, especially in the first stage

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of the problem-solving process,

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to facilitate understanding.

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Successful problem solvers are able to translate

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and integrate information in the problem

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to generate a coherent mental representation

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that they then use to plan to solve the problem.

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There are also many uses or purposes

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for visual representations.

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They include summarizing and organizing information,

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making coherent the abstract relationships,

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reducing working memory demands, and also reasoning

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about the story situations.

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When visual representations are used appropriately,

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they can lead to deep understanding of the problem,

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and that in turn makes possible transfer of learning

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to new problems and increases retention.

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I would like to take an example

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from the Practice Guide and illustrate how different types

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of visual representations can be used.

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In this problem, during a sale,

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prices were marked on down by 20 percent.

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The sale price of an item is $84.

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What was the original price of the item before the discount?

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We are using the percent bar model,

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we can organize information in the problem

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and make concrete the relationship

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between the original, the decreased,

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and the final amount.

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This model clearly represents the information described

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in the problem.

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We can use the relationships depicted

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in the visual representation

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and convert it into an equation.

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In this representation, the translation

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from the visual representation to an equation is direct.

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Using the same problem,

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this time let's use the strip diagram

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to show how we can represent the information

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in the problem.

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It is important to understand that 20 percent is the same

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as twenty-hundredth or one-fifth of the whole.

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So the strip diagram here shows how the original amount

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is divided into five equal parts and one

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of the five parts represents the discount amount and four

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of these five parts represents the sale price.

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From the strip diagram,

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we can get the equation 4/5 of X equals 84.

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Visual representations such as schematic diagrams,

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strip diagrams, tables, and percent models

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that show relationships described

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in the problem are much more powerful tools than pictorial

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or iconic representations that focus attention

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on the surface details, or the physical appearance,

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of elements described in the problem.

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For struggling students,

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who may have cognitive working memory deficits,

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providing them with a visual representation can enhance

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problem solving because it would reduce the working

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memory demands and allow them to focus

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on the problem-solving process.

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Although, there is evidence

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that these students can benefit or can be taught

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to use visual representations to enhance their problem-

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solving performance, encouraging them

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to just visualize the problem

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or draw a diagram may not be very effective.

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These students' difficulty in problem solving is related

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to the representation used.

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The representation needs to be understood in order

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for it to be useful.

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Selecting out an appropriate visual representation is

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not easy.

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One consideration is to select a visual representation

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that appropriately links it

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to the underlying mathematical concept.

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Area models and number line diagrams can be excellent

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models for improving student understanding of fractions

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and formal computational procedures.

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Schematic diagrams, strip diagrams, tables,

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percent bar models are excellent for understanding

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and solving ratio proportion and percent problems.

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What is important is that we need to make sure

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that the visuals provide an effective means

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of visualizing the problem to facilitate problem solving

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or understanding of the problem.

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When the goal is to use external visual models,

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then instruction needs to focus

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on helping students identify

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and represent the relationships described

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in the problem on to a diagram.

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When the goal is to have students generate a diagram,

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instruction needs to be multifaceted and focused

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on understanding of diagrams and their purpose,

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how to generate a diagram, and use diagrams

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as tools to reason with.

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Having students explain their use of a visual representation

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or how quantities in the problem are represented

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in the diagram is useful

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in understanding students' misconceptions

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that would then allow us to identify

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and address any learning problems during instruction.

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[Music]