When I first read about concrete examples, my first thought was manipulatives. It has always been recommended students use manipulatives to help them learn mathematical concepts but does it really do this? Do manipulatives actually help students transfer knowledge?
Well, lets see! It was discovered that students may not picture the concept the same way the teachers do who already know the concept. Students may not interpret the action in the way the teacher hopes and they may not carry the meaning of the mathematical idea. Unfortunately, students are often taught to use manipulatives in a rote manner.
Manipulatives work best if reflect on their actions with manipulatives. This helps them build meaning. This is integrated concrete knowledge which combines separate ideas into an interconnected web of knowledge. In other words, mathematical concepts become integrated concrete by how well they are connected to other ideas and situations.
It does not matter whether the manipulatives are virtual or real, the students who use them need to build the ideas from the manipulation of the representations and thinking about their actions.
The one idea that comes up again and again in literature is that giving the students the manipulatives does not guarantee they will understand the concept. It requires careful planning and good transitions. The teacher needs to remember the purpose of the lesson when using manipulatives and must create a carefully thought out sequence of steps.
It was found that the higher the level of guidance the teacher provides, the better the learning outcomes in retention and in problem solving. When they reach a point where they require a lower level of guidance, their are better able to transfer their knowledge.
So all in all, if we use manipulatives in math, we need to make sure that we carefully plan their use so the students gain the knowledge we want. If we do not, then we may just be using them in a rote manner so students may not learn. I'd love to hear about your experiences.