I’m working on a couple of short videos comparing the standard algorithm for a multiplication and addition, and considering some ways of using other algorithms which are more likely to make sense for students.
To be clear, these presentations do not enough justice to student-created algorithms, which I would strongly recommend as a starting place for exploration of algorithmic reasoning. These two videos are merely an attempt to compare two different algorithms against each other, and to provide some support for teachers interested in learning more about why we might want to use something other than the standard algorithms in our classrooms.
I also want to make it clear that how I am describing these distinctions in the videos is not how I would introduce them to children (or adults for that matter, if I have the time). Instead, I would start with setting up realistic situations, whole group, and small group discussions around investigating these operations, using both manipulatives and symbols to represent numbers in the algorithms. A really useful activity, for example, is for children to be making these comparisons themselves, so that they can look for patterns in different operations, and abstract these patterns into general rules they can follow to make their use of any algorithm easier.
I see these algorithms as a useful way to get started with abstract reasoning, provided they are framed in a certain way, as described below.