I’m reading Dylan Wiliam’s "Embedded Formative Assessment" book (which I highly recommend) and this paragraph jumped out at me:

"To illustrate this, I often ask teachers to write 4x and 4½. I then ask them what the mathematical operation is between the 4 and the x, which most realize is multiplication. I then ask what the mathematical operation is between the 4 and the ½, which is, of course, addition. I then ask whether any of them had previously noticed this inconsistency in mathematical notation — that when numbers are next to each other, sometimes it means multiply, sometimes it means add, some times it means something completely different, as when we write a two-digit number like 43. Most teachers have never noticed this inconsistency, which presumably is how they were able to be successful at school. The student who worries about this and asks the teacher why mathematical notation is inconsistent in this regard may be told not to ask stupid questions, even though this is a rather intelligent question and displays exactly the kind of curiousity that might be useful for a mathematician — but he has to get through school first!" ~ Dylan Wiliam, Embedded Formative Assessment, 2011, p53

Mathematical notation has been developing since the introduction of writing and has largely grown organically with new notation added as it is needed. In fact, if a mathematical concept is developed in different cultures, it is entirely likely that each culture will develop its own mathematical notation to describe the concept, and these mathematical notations inevitably end up competing with each other, sometimes for centuries.

This observation by Dylan Wiliam suggests to me that difficulties in mathematics for some students are almost certainly related to the notation that we use to represent it (especially in classrooms where mathematics is largely presented to students in completed form, rather than being constructed with students), and that people who end up good at math in school may be good at being able to switch meaning based on context.

Can you think of any other examples of mathematical notation which are potentially inconsistent with other mathematical notation? I’ll add one to get the list going:  which is clearly inconsistent with algebraic notation, and potentially with fractions too.