Having a notion of what’s inside our students’ heads can help us provide instruction that connects old knowledge with new, informal sensibilities with formal understanding. Educators sometimes use the swiss cheese metaphor when describing student knowledge. It’s full of holes. We simply need to find the holes and plug them. Unfortunately, this plug-the-holes approach only works if the new material bonds with what’s already in place. Otherwise, the plug just falls out.
Sometimes building on to the existing knowledge base will work, but it’s difficult to know what you’re building onto with every child. Instead, it can be more efficient and effective in the long-term to rebuild the structure from scratch. Provide students with the common concrete experiences that can provide the tangible foundation out of which you abstract the desired mathematical concepts. In any case the connection is key. If students see the knowledge as isolated bits of information to remember, you’ve got a cognitive overload situation on your hands. And you don’t want that.