I have often heard from teachers the following, “My students can do the problems with counters, but when I take the counters away, they can’t do the problems anymore.” I have seen many math textbooks that introduce addition equations with counters in early grades and after a few times using the counters, the students are asked to do the problems with the numbers only and no counters. Even if several days of using counters is part of the sequence of instruction, there still comes a time when the teacher is instructed to put the counters away and students are left to figure out the problems with numerals alone. So why do so many students struggle at this point?
The research we have been looking at suggests a reason for this. In between using the counters and only using the numerals, there is a time that students need to imagine using the counters. The gap between the manipulative and the abstract symbol is simply too great for students to traverse in a solo leap. Instead they need time to first think about what the counters mean and then to have time to do problems in which they imagine using the counters. So what does that look like? How do we know they have thought about the counters enough? How do we know they have imagined the counters enough? All of these questions are answered by research that we at the AIMS Center are coming to understand and that will be part of upcoming blogs.