I have written a fair amount on visual representations of abstract physical quantities such as energy, power, current and momentum. I particularly favour the bar model and the box model. I was initially drawn to them because I believe they reduce the cognitive load of abstract mathematics and because you can make use of dual-coding.
I learnt about bar-model through observing primary and secondary maths teachers – and all of the educational research comes from maths. There are two important reports which refer to visual and physical representations of abstract concepts in maths:
- Improving Mathematics in Key Stages 2 and 3 by the Education Endowment Foundation (EEF) and
- Improving Mathematical Problem Solving in Grades 4 Through 8 by the What Works Clearinghouse.
My aim in this blog is to take the evidence for generalised maths problem solving and apply it to science.
Using visual representations to clarify problems
Visual representations clarify problems. By representing the problem as a simple quantity, length or area, the learner is able to declutter.
| “Using visual representations prior to introducing equations brings several benefits to students, helping them organize the information in a problem, distinguish relevant from irrelevant information, clarify the goal of the problem, see relationships, and then focus on mathematical reasoning.”
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Developing a deeper understanding of a problem
A visual representation allows the abstract quantity to be treated as a more common place quantity, such as length or area. This acts as a bridge to deeper understanding of the concept involved and how it behaves.
| Research suggests that students who develop visual representations prior to working with equations are more effective problem solvers. This may be because visual representations help students develop a deeper understanding of the problems they are working with. The right type of representation can help a student get a coherent view of the problem by identifying and organizing pieces of relevant mathematical information. Specifically, the visualization helps students summarize what key information is known and see what the problem is asking them to solve for. Use of an appropriate visual can also reveal the relationships between the quantities identified in the problem. Once students grasp these relationships, they can focus their attention on mathematics reasoning and the problem-solving process. Students are also in a better position to express a problem using equations.
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How many visual representations should you use?
Don’t go crazy with the visuals – if you can reuse one visual in multiple contexts (e.g. bar-model) your learners will benefit more.
Familiarity with powerful visualsTeachers are advised to consistently use a few powerful visual representations. A powerful model or representation is one that has a variety of applications, such as a number line or strip diagram. If students work with a particular visual representation when they encounter a certain type of problem, they are more likely to grow comfortable with that tool and use it on their own. Students are also then less likely to use narrative pictures, which can distract them from the essential mathematical information in the problem.
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How should I use the visual representation?
The visual allows the teacher to talk through her thinking. She can point at the abstract concept (energy, momentum, charge etc) and describe what is happening to it. If you are using a bar-model, talk through what the length of each segment represents. If using a box-model, what are the axes?
Talking through the visualTo help students learn how to employ visual representations, teachers can talk aloud about what they are thinking and the decisions they are making as they reason through a problem. During the think-aloud, teachers should demonstrate how they identify what information will be placed in the diagram and what aspects of the problem are irrelevant. It is essential that teachers explain why the representation they are using is appropriate for the problem at hand. It is just as important for students to explain to the teacher how they are setting up a diagram and representing the quantities in the problem. By listening to their students’ reasoning, teachers are better able to identify and address possible misconceptions. As teachers grow to understand how their students are thinking about problems, they will be able to introduce them to the idea that there is more than one way to think about a problem.
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I hope that’s useful. As always, I’m grateful to the primary and secondary maths teachers who’ve supported me on my bar-model journey.
Ben
Reading for Learning 