Start with a recap of Hooke's law. As a visual aid add 1 N weights (i.e. 100 g masses) to a suitable spring. If you also have a compression spring then so much the better. It should be apparent that
extension (or compression) Dx ? load
(Note: although many text books simply use x for extension when discussing Hooke's Law, it is helpful to use Dx to avoid confusion later with the original length x when discussing the Young Modulus.)
Note that a pair of forces are involved when applying a tension or compression. If only one force acted on the sample, it would be ?unbalanced?, which implies the sample would accelerate. If you suspend a spring from a support and hang a weight from it, the weight is one force; the other is the upward force provided by the support.
Stiffness k obviously depends upon the actual material; it also depends on the dimensions of the sample. Thicker samples stretch less per newton than thinner ones. Imagine two identical samples in parallel ? twice the cross-sectional area A implies half the extension for given load, as each sample supports half the load, so k for the system as a whole is doubled.
What about two identical samples in series? Both are subjected to the same load, thus each stretches the by the same amount, so the total stretch is doubled, so the stiffness for the system as a whole is halved.
Check the above conclusions in a demonstration using (identical) springs in parallel and in series.