Continuity and Discontinuity
Let's go back to that mountain that you want to climb. You're standing on the very bottom of the mountain, and you want to get to the top. But when you look closer, you notice that there are two ways up. On one side of the mountain is a path that involves walking uphill until you get to the peak. On the other side, someone has carved stairs into the side of the mountain so that you can climb up to the peak that way.
The path is a lot like the continuity view of development. Proponents of the continuity view say that development is a continuous process that is gradual and cumulative. For example, a child learns to crawl, and then to stand and then to walk. They are gradually learning how to walk. It's just like hiking up the mountain path: a slow, steady ascent that leads to the top.
On the other hand, some people see development as consisting of different stages. The discontinuity view of development believes that people pass through stages of life that are qualitatively different from each other. For example, children go from only being able to think in very literal terms to being able to think abstractly. They have moved into the 'abstract thinking' phase of their lives. As you can imagine, discontinuous development is like walking up the stairs: a series of stages, or steps, that get you to the top of the mountain.