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.
Dave says:
Without a doubt, the -1 superscript for inverse functions AND reciprocals confuses all kinds of kids. It doesn’t matter how many times I remind them, they mess it up, and it’s really not their fault. It’s a dumb way to mix things up.
July 9, 2013 — 9:36 am
David Richeson says:
Off the top of my head, two examples of notation that often confuse some students:
Inverse trig function notation sin^(-1)(x)≠(sin x)^(-1) even though sin^2(x)=(sin x)^2
Function notation in general: sin(x) confused with “sin” times “x”
July 9, 2013 — 9:44 am
@dazmck says:
Agree that this is all confusing. How I wish we’d gone for square brackets for functions – f[x] instead of f(x), keeping regular parentheses for multiplication. Oh well. It at least helps to be aware of the problem!
July 9, 2013 — 9:50 am
Oskar says:
But it means the same thing. It is quite useful (and a key to success in mathematics) to identify a scalar $a$ with the function $x\mapsto ax$. The problem is that most people (prefer to) confuse a whole function $f$ with its action on the variable $f(x)$, so when it comes to inversion (with respect to function composition) to be distinguished from inversion (with respect to algebraic multiplication) many of us will write $f^{-1}(x)$ to indicate the much simply written $1/f(x)$. On the other hand, if strictly necessary one may always pull out the need symbology, and write $f^{-1}_\circ$ to emphasise which operation we are inverting against. As for action of $f$ on the variable $x$, I think the best notation was invented by computer programmers as $x\dot f$.
June 6, 2018 — 10:08 am
Donna says:
The Casio Classpad calculators trip kids up. They DO treat 4 1/2 typed as a mixed number using the fraction template for the 1/2 as 4 x 1/2. I regularly need to stress the need for improper fractions. Guess the programmers didn’t think about this inconsistency.
July 9, 2013 — 6:38 pm
Rogan says:
Regarding the initial issue, I generally explain that product notion in algebra is there to remove confusion, since x is widely used and looks very similar to the multiplication symbol. Hence we avoid 4xx. It is also consistent with language, where we say ‘4 eggs’, rather than ‘4 times egg’ or ‘4 lots of egg’
One notation my students struggle with is ‘log’. As with David Richeson’s example of ‘sin’ above, students sometimes treat it as a variable, and try to divide by ‘log’. My time machine solution would be to replace ‘log’ with a stylised ‘L’ which encloses the argument – exactly like the stylised ‘r’ for square roots. No student has ever tried to divide by the ‘r’ to undo a square root.
July 9, 2013 — 6:58 pm
david land says:
In handwritten algebra two back-to-back touching curves make the variable called x, but the times sign is pure St Andrew’s cross drawn with straight lines not curves. It is only through the inadeqacies of keyboards and the resultant slackness of some educators that there is any ambiguity in the notation for such expressions as 3xy or 3xx or xx3 etc.
October 11, 2018 — 3:07 pm
Chris Sorrentino says:
Hi David, great observation!!
I’ve noticed that another tough obstacle is understanding the meaning of – (minus) sign: when relative integer numbers are introduced, they mistake the use of minus as negative sign of relative numbers for its meaning as a sign of operation linking two natural numbers.
I think that the confusion is linked to the omission of + sign at the beginning of an expression.
E.g.:
If you write 3-2, it can be understood both as
1) 3 – 2
or
2) +3 -2
This apparently has no concrete consequence in numerical terms, but the meaning is different and in longer arithmetical expression it can lead to mistakes.
What do you think about it?
Chris
July 10, 2013 — 9:00 am
Rachel says:
Chris–the minus sign is even more confusing, because it can be used as:
a minus sign [5 – 3]
as a negative [-17]
and as an opposite sign [-x or -(-13)]
While they are all related, the differences are also pretty nuanced and, I think, hard to articulate. I (and many other math teachers) are so familiar with these meanings we don’t always realize when the meaning/intention switches. So it is even harder to try to explain this to students–I don’t always catch on to their confusion!
July 11, 2013 — 8:53 pm
Lura says:
Thank you sincerely for posting about this. I came across Dylan Wiliam’s work in a class I am taking and googled the phrase “inherent ambiguities” with his name and found your post. This excerpt explains it more clearly than the one I read. Your post has helped me to articulate my own difficulties with Math. So, thanks!
July 14, 2013 — 9:21 pm
David Wees says:
I’m glad I could help, Lura. Good luck with your class. Dylan Wiliam’s work is worth the effort to understand.
July 14, 2013 — 10:33 pm
Guest says:
Another ambiguity: misuse of the equals symbol, where it is used to indicate an equality, but the equality expression itself is treated as having the value of the value being compared, rather than a value of ‘true’.
September 3, 2014 — 10:27 am
Ricky says:
For me, one the most confusing: The Pipe |.
Various meanings:
– in some cultures 1 ( handwritten )
– ‘divides’ relation
– seperator between a formula and its conditions
– absolute value
why don’t the ‘big mathmaticians’ solve it? >.<
June 18, 2015 — 10:46 am
Bonnie Dunleavy says:
If only we had more teachers not just willing, but able in respect of time and space, to truly investigate each child’s honest intelligence and celebrate it via the medium of SUCCESS! Many of us have brains whose chemical makeup causes us to struggle in regard to this nonsensical notation principle. Many of us, even more so when it comes to fractions!!! I suppose we naturally think more logically. Super interesting!
December 21, 2018 — 4:59 am
Paul says:
We have had a brutal fight with friends on this simple mathematical problem: 6:2(1+2).
One says the results is 9 whilst others say the result is 1.
It all comes down to mathematical notation and order of calculations.
Ultimately the question is what is the correct interpretation of notation.
6:2(1+2) can be represented as 6/1 / 2(1+2) and in this case the result is 1.
Others claim the division shall come first so the result is 9.
I’ve entered the problem into Wolframalpha project and when entering it as 6:2(1+2) the result is 1. When entering it as 6/2(1+2) the result is 9.
So the question is what is right.
January 27, 2019 — 1:26 pm
David Wees says:
This is a case, as I understand it, of the notation changing meaning over the years. Historically in the expression, 6÷2(1+2), 2(1+2) would be grouped together and calculated first, resulting in the answer of 1. However, implicit grouping has fallen by the wayside and so the standard order of operations holds. (1+2) = 3. So this is the same as 6÷2×3. Since multiplication and division have the same precedence, this is calculated left to right, hence it becomes 3×3 or 9. So in some sense, both sides in your argument are correct.
January 28, 2019 — 3:23 pm
Mike says:
Regarding 6 ÷ 2(1+2)
If this were written using a variable a=1+2
6 ÷ 2a, I would suggest most people would say “Six divided by two ‘a’ “ and the answer is 1.
If the original equation had been written as 6 ÷ 2 x (1+2) then it becomes 6 ÷ 2 x a and the answer, most would say, is “Six divided by two, times by ‘a’ “ and the answer is 9.
To avoid the ambiguity, more parentheses should be used eg
(6 ÷ 2)a= 9, when a = 1+2
6 ÷ (2a) = 1, when a = 1+2
In reality the order of operations rules need to account for the use of a number in front parentheses such that replacing the parentheses with a variable doesn’t change how the equation is read.
July 17, 2021 — 3:39 pm