Many of the difficulties that students have with mathematical symbolism seem to stem from the use of a horizontal form for an equation and a vertical form for computation. The differences between these two forms are discussed. (JT)

Some students can perform operations on fractions when using physical materials, but are unable to transfer this ability to operations with symbolic expressions. They seem to view the manipulative and paper-and-pencil problems as different and are not surprised to get different answers. (SD)

If the principles of languages are applied to the learning of arithmetic, it appears that elementary grade pupils are being introduced to mathematical symbols at too early an age. (MP)