For pre-service teachers, projects can be given to develop lesson plans to use commutators to teach students to understand common errors.
Distributive Law Error
a*(x+y) = ax
Define a commutator
a*(x+y) – ax = ay
We can call this a fallacy distributive commutator.
One definition might be
[a,x,y] = a*(x+y) – ax = ay
Using this definition, we have that
[a,y,x] = ax
[a,x,y] + [a,y,x] = ax + ay = a*(x+y)
We might write this as
[a,(x,y)] = a*(x+y) – ax = ay
Project: Identify common errors with fractions. Show how using the fraction commutator [p,q] = p/q – q/p can identify these errors. Develop a lesson plan for students to compute these commutators.
Project: In algebra, a student might think a/(b+x) = a/b or a/x. Develop a commutator for this error. What are its properties?