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Visible Learning for Mathematics

Visible Learning for Mathematics

Effective teaching is the goal of every teacher of mathematics. How do you know that your teaching is effective? What strategies should you choose? What tasks?

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Home > Topdrawer > Fractions > Misunderstandings > Using rules blindly

Using rules blindly

Some students find it difficult to know which procedure to apply in a particular situation. A lack of understanding of how and why a procedure works is usually the underlying cause.

There appear to be three common pathways to students establishing a pattern of applying a procedure inappropriately. These are:

  • learning a useful and effective strategy for one type of task, then not realising that when the task changes, the strategy must also change (e.g. comparing unit fractions by just comparing the denominators, then applying the same strategy to comparing non-unit fractions)
  • encountering an unfamiliar situation and drawing on a familiar strategy that seems to have some connection to the new situation (e.g. trying to add fractions by treating the numerators and denominators as whole numbers, and adding them)
  • being taught to use a new procedure, and allowing it to supersede all previous strategies (e.g. when finding a common denominator, always multiplying the two denominators, even when there is a much simpler relationship between the denominators such as one being a multiple of the other).

Unit and non-unit fractions

A common misconception is that the rule for comparing the relative size of unit fractions can also be used for comparing non-unit fractions.

Strategy testing

To avoid misuse of old strategies in new situations, teachers should prompt students to examine and evaluate their strategies, and deliberately support the development of new ones.

Sense of size

One way to reduce the likelihood that students will blindly apply inappropriate procedures is to strengthen their sense of the size of fractions, particularly in relation to the whole.

Comparing non-unit fractions

The digital learning object provides a tool to help students develop strategies for comparing non-unit fractions.