There's no conflict between your high school teacher's advice
To prove equality of an equation; you start on one side and manipulate it algebraically until it is equal to the other side.
and your professor's
To prove a statement is true, you must not use what you are trying to prove.
As in Siddarth Venu's answer, if you prove $a = c$ and $b = c$ ("working from both sides"), then $a = c = b$ by transitivity of equality. This conforms to both your teacher's and professor's advice.
Both your high school teacher and university professor are steering you away from "two-column proofs" of the type:
\begin{align*}
-1 &= 1 &&\text{To be shown;} \\
(-1)^{2} &= (1)^{2} && \text{Square both sides;} \\
1 &= 1 && \text{True statement. Therefore $-1 = 1$.}
\end{align*}
Here, you assume what you want to prove, deduce a true statement, and assert that the original assumption was true. This is bad logic for at least two glaring reasons:
If you assume $-1 = 1$, there's no need to prove $-1 = 1$.
Logically, if $P$ denotes the statement "$-1 = 1$" and $Q$ denotes "$1 = 1$", the preceding argument shows "$P$ implies $Q$ and $Q$ is true", which does not eliminate the possibility "$P$ is false".
What you can do logically is start ("provisionally", on scratch paper) with the statement $P$ you're trying to prove and perform logically reversible operations on both sides until you reach a true statement $Q$. A proof can then be constructed by starting from $Q$ and working backward until you reach $P$. Often times, the backward argument can be formulated as a sequence of equalities, conforming to your teacher's advice. (Note that in the initial phase of seeking a proof, you aren't bound by anything: You can make inspired guesses, additional assumptions, and the like. Only when you write up a final proof must you be careful to assume no more than is given, and to make logically-valid deductions.)
Math has the advantage that we can actually accomplish things in the "comments"
(that is, in the narrative passages between formulas).
Math also tends toward notations that are compact rather than self-documenting, perhaps because the "documentation" is already so baked into mathematical writing styles.
There are also some relatively firm mathematical conventions for subscripts and function parameters, one of which is that there is a first subscript/parameter, a second subscript/parameter, etc.
So if (for example), $i=2,t=1,k=3$, then $x_{i,t,k}$ is not generally the same thing as $x_{k,i,t}$,
unless it happens in the case in hand that $x_{2,1,3} = x_{3,2,1}.$
Function parameters, for example $f(i,t,k),$ follow similar rules. Position is everything. The parameters don't generally need names.
In contrast, functions in software programs often have too many parameters with no obvious sequence to help you remember which is which. They often have parameters you use only sometimes; the rest of the time those parameters take default values implicitly. These are strong motivations for using named parameters.
On the other hand, there is a mathematical notation that is something like a named parameter. For example, suppose we define
$$ F = 3x^2 + y^3. $$
Then we could write
$$ F \bigr\rvert_{x=2,y=4} = 3(2^2) + 4^3. $$
If you want to read the left-hand side aloud, you can say,
"$F$ evaluated at $x=2,y=4$."
There are some other mathematical conventions that rely on similar ideas. For example, every time you see someone write a partial derivative in a form like
$$ \frac{\partial f}{\partial x}, $$
they are using $x$ as a named parameter, that is, the implicit assumption is that $x$ represents one of the parameters of the $n$-parameter function $f$ and that (at least in principle) we know which of those parameters it represents (for example, the first parameter).
But I have noticed that the $\frac{\partial f}{\partial x}$ causes many people to have difficulty understanding how partial derivatives actually work. (See this question for some more discussion.)
When you start naming variables in math the results often aren't particularly happy. This may help explain why it doesn't happen more often.
But if you look long enough I am sure you will find some mathematical work somewhere in which named parameters are used explicitly in a way similar to the named parameters of a programming language,
in some case where a function has many parameters or where it is hard to remember what the first parameter means as opposed to what the second parameter means. Mathematical notation is extremely flexible (as long as you tell the reader what you're doing when you introduce something new) and can be redefined to suit any needs, unlike almost any programming language.
Best Answer
In mathematics exists something called 'abuse of notation', where you omit something when you know everyone reading it knows what you are meaning (or it is just impossible to have a crystal clear notation as in many diagramms in homological algebra), or we use the same notation for different objects. Again normally everyone knows what is meant, but it can be inconvenient for students new in the subject.
At the question with the function notation. Basically you see it correctly.
There are different notations which are used. For a function $f:X\to Y$ you can write $x\mapsto f(x)$, or $f(x)=y$ and this would mean exactly the same.
When first introduced to functions one would note it like $y=x$ or $y=x^2$. This notation has it flaws.
First of all the f(x)-notation can contain more information. When we have $f(x)=x^2$ then $f(-1)=1$ and $f(1)=1$. So this notation gives us the point we are talking about (1|1) or (-1|1) while the $y=\dotso$ notation does not remember the point $x$, and only its value. When first learning functions we create these value tables, where we keep track of $x$ and $y$.
The notation $y=\dotso$ should be viewed from a purley didactic point of view. Making clear how to get an y-value and so on. That these value tables are important, and how to scetch a graph. As many problems at this level have to be solved by putting a given point in the equation. Like calculate $m$ and $b$ in $y=mx+b$, when you have points given. From the coordinate names it is clear where to put them.
Later when the student already is familiar with this, we can move on to a more fitting notation, which might be difficult for younger people. When we care about other questions. What is the derivative, how to get extremum and so on. However, students that learn the f(x) notation often do not really understand it. They know what to do, but they would talk about the function f(x) not understanding that this is supposed to note a value on the y-axis.
Also do not treat this equation correctly. One would see calculations like
$$f(x)=2x^2+4x+2 |:2$$
$$f(x)=x^2+2x+1 $$
when correct would be $\frac{f(x)}{2}=x^2+2x+1$, what comes from that most of the time the questions these equations have to answer is when $f(x)=0$, so it does not matter.
Just my two cents.