A single symbol or notation means many things - does this actually cause problems for learners?

Question

This is a somewhat abstract and ill-formed question, but I hope the examples clarify it. The question concerns a common pattern in didactic literature.

For example:

1. The equal sign has several different meanings, like assigning a value and comparing things. The notation for these is the same.
2. The minus sign has several different meanings, like subtraction, the sign of a number and multiplication by minus one.
3. Sometimes this is more abstract, such as in so-called abuses of notation, where $x^3$ is a function and we do measure theory on it, rather than the equivalence class. And we just gave a formula, not a function, which requires specifying where the function is defined, where it maps to, and which letter is the variable there.

The common factor in these situations is that notation hides complexity or distinctions. A thing I see in didactic literature is teasing these apart, which is all well and good.

The question: do these situations where notation hides complexity cause problems for learners?

An alternative hypothesis is that pupils and students find, say, the equal sign difficult for some other reason, which is not really related to this hidden complexity. It is known, for example, that pupils mostly see the equal sign in tasks such as 3 + 18 = ? , so they learn it means that "the result or next step in the calculation is". But seeing a thing often and thus learning the most obvious usage is a different reason for difficulties than the concept itself having hidden complexity, for example the assignment and the comparison. Note that "the result or next step in the calculation is" is not really mathematically accepted definition of the equality sign. Some kind of arrow might sometimes be used for this purpose, at least on blackboard work, but not really the equal sign.

Or maybe they are the same thing. This is what I am asking for.

A positive answer could refer to research that shows how many misconceptions are related to fundamental structures hidden by notation. A negative answer would show that most misconceptions have their root in some other phenomenon than the concept in question having many different and hidden meanings.

Answer 1

I don't have any research references handy, though I'm sure many exist for this question (particularly for the difference between a function and an expression in writing things like $x^3+1$). However, one I can vouch very clearly causes trouble in my own experience is the use of parentheses in American mathematics education.

• Parentheses mean grouping, which may or may not be associative: $2^{\left(2^3\right)}\neq \left(2^2\right)^3$
• Parentheses mean functions: $f(x)=x^2+1$
• Parentheses mean multiplication: $2(3)=6$

Unfortunately, especially the latter two are often confused - often enough that the "If I had a dollar for each time ..." cliche might actually hold true. Why? Some examples of miscomprehension I come across every year:

• If $f(x)=3x$ then $f=3$
• If $g(5)=30$ then $g(10)=60$, even if $g(x)=x^2+5$
• If $\ln(x)$ is the notation then $\ln(x+y)=\ln(x)+\ln(y)$ must be true.
• A related issue is pronunciation of "f of x" as "f x" or even "f times x".

I'm not saying these problems couldn't show up in other ways, but the implicit linearity one gets from the similarity of these notations is pretty aggravating, as an educator, and surely frustrating to students who are tripped up by it. The function concept is hard enough (note that it took Euler to first use it sort of properly) without having its notation shared by something else important.

---

Side note: My understanding is that in a lot of the Commonwealth/UK the last meaning of parentheses (brackets in the UK?) is not anywhere near as common, writing instead $2.3$ or something. I'd love to see precise evidence of when the parenthesis convention became popular in the United States (and perhaps elsewhere).