In my last blog I discussed the gap between using manipulatives and using numerals only as being a gap too large to traverse in one step. What then bridges the gap? Research that has been the focus of study at the AIMS Center suggests that the answer is to be found in the mathematical knowledge of the child or in other words, what the child thinks. Children need to learn to imagine the manipulatives when they are not there until the numeral for, say five, has enough experiences associated with it to have meaning that allows the child to work with only the numeral.
Think of a time when you went to the beach. Think about walking in the sand and maybe allowing the surf to lap up against your feet. Think of the smell of the ocean and the cry of seagulls overhead. As you do this, if you can, you draw from a first hand experience. That experience is like the manipulatives. As you remember it, you relive the experience over and over again without having to be at the beach. This allows you to work with the experience in your mind.
For example, I can use the remembered experiences of going to the beach to understand other situations. I have never been to Hawaii, but I can imagine the beaches there, based on my many experiences in literally dozens of beach experiences I have had around the world. It allows me to imagine the beach without having it present. Imagining the beach also allows me to change my experiences at the beach. I can imagine sitting on the sand talking with a friend, even if I have never done that with that friend. I can perform mental operations on the experience because I can bring it up in my mind and work on it in my mind. This is what we want students to be able to do.
Students have to be put in situations in which they are encouraged to imagine the manipulatives when the manipulatives are not present. To do this they need lots of time with the manipulatives in which they think about them. You can’t drive by the beach one time and have a very rich experience to remember. Students need to be put in situations in which they have to remember the manipulatives. Instead of leaving both counted sets of blocks viewable to find the total, consider hiding the second addend. You can hide the blocks after they count them first. This encourages the students to remember the blocks in order to count them. They will imagine the blocks and count the spots on whatever covers the blocks as though the blocks were there. They might use fingers to stand in for the blocks. This is the first step in being able to traverse the gap between the counters and the numerals, and it is the step that is often forgotten.
In my next blog we will continue building the bridge to traverse the gap between counters and numerals.