What is it about two line segments meeting at a point which makes them produce an angle?
What is it about 3x + 4 = 10 which makes it an example of a linear equation in one unknown? What can be changed without losing this property?
You don’t understand a concept unless you recognise examples of it in a variety of contexts. Actually you need more. You need to appreciate what it is that makes something an example, and this means knowing what aspects or features can change and still it remains an example, and how to construct other examples.
For example: line segments form an angle when there one of the segments is considered to be the initial arm, and the other the final arm, and the angle is the amount of rotation to get from the initial to the final arm. Of course the segments can change in length (other than zero, perhaps), and the whole image can be anywhere in any orientation in the plane.
For example: a linear equation can be formed by equating two expressions, each formed by … it is actually quite difficult to articulate a definition, yet learners are expected to recognise linear equations in one unknown or variable, and to be able to solve them. They do this by becoming implicitly aware of the features which can change and of how you can alter an equation to produce a more comoplicated one. [see
Simultaneous Equation Obstacles]
Ference Marton proposed that learning is about becoming sensitised to possible variation [see
variation theory]. Anne Watson & John Mason proposed that to appreciate a mathematical concept it is important to have access to a rich space of examples [see
example space]. This means having access to a variety of examples and to construction techniques for creating more.
When providing learners with examples, it is important to make sure that there is a range of examples with variation in the important features, sufficiently close together than learners experience what it is that it is possible to change.
It is also important to recognise over what range that change is actually permitted. For example, are the numbers involved permitted to be whole numbers, integers, fractions, decimals? This is referred to as the range of permissible change.
Being aware of dimensions of possible variation and their corresponding ranges of permissible change is important as a teacher because very often learners are unaware of some important dimensions, and have only a restricted sense of the permissible change.