Skip to main content

Questions tagged [notation]

For questions about good use of notation, comparison of specific notation, motivation of notation.

Filter by
Sorted by
Tagged with
81 votes
6 answers
8k views

Issues with "equals", where does this come from and how do I combat it?

An issue I see with students a lot is abuse of the equals sign. For example, one problem asked "what is the degree of the polynomial: $\text{polynomial}$?", and I got answers like "$\text{polynomial}=...
user avatar
45 votes
21 answers
7k views

How to help new students accept function notation

I am struggling to help some of my new precalculus students accept function notation -- something new to them this term. I am looking for strategies to help them adopt this new notation. Their main ...
Nick C's user avatar
  • 9,639
39 votes
11 answers
3k views

Reasons for (not) distinguishing $f$ from $f(x)$

Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. In teaching at school and university, notation is quite often mixed up, e.g. the function is ...
Anschewski's user avatar
  • 4,811
35 votes
1 answer
2k views

Metonymy in mathematics

Metonymy is a figure of speech where a word or another expression is used to mean something other than its literal meaning. This phenomenon is not restricted to the "usual human languages" (such as ...
Joonas Ilmavirta's user avatar
32 votes
10 answers
2k views

Should students be asked to use more than one notation for the derivative in an introductory calculus class?

There are many, many ways of writing the derivative of a function $y=f(x)$: $$\frac{d}{dx}y, \frac{dy}{dx},\frac{d}{dx}f(x), \frac{df}{dx}, \dot y, D_x f,f',y',f'(x),f_x$$ and so on. Students often ...
Brian Rushton's user avatar
32 votes
6 answers
2k views

What is the rationale for the absent (+) in mixed fractions?

Why are students taught to represent one and a half as $1 \frac{1}{2}$ rather than $1 + \frac{1}{2}$? This mode of expression seems standard at least throughout North America. I believe that it is bad ...
NiloCK's user avatar
  • 5,020
32 votes
6 answers
3k views

Allowing nonstandard mathematical language and/or notation

How important is enforcing standard mathematical language and/or notation? Today, I was questioned by a writing instructor as to how vital it is to correct students when they explain something using ...
Nick C's user avatar
  • 9,639
29 votes
6 answers
8k views

Misuse of parentheses for multiplication

I'd like to raise the issue of constant misuse of parentheses in the U.S., and I'm wondering if anybody else shares the same feelings, has had the same issues, knows any history behind it, and can ...
zipirovich's user avatar
27 votes
6 answers
11k views

Should we stop differentiating between ln and log?

In many U.S. middle schools and high schools, $\ln$ and $\log$ are treated differently, with the intent that $\log$ is equivalent to $\log_{10}$. However, in undergraduate courses and in the academic ...
Nahmid's user avatar
  • 403
25 votes
11 answers
6k views

Why do we still teach the determinant formula for cross product? And is it as bad as I think it is?

The cross product is an important vector operation in in any serious multivariable calculus course. In most textbooks that I'm aware of, right after the definition, we always introduce the ...
user avatar
24 votes
5 answers
6k views

Correcting how a student writes symbols

One of my college students writes the Greek letter $\pi$ as a script n with a bar over it, like $\bar{n}$. [There is actual space between the letter and the bar.] I have never seen this before, and ...
Nick C's user avatar
  • 9,639
24 votes
7 answers
5k views

Repeated addition: standard notation?

My daughter showed me the picture below, which came from 9GAG. It shows a question on an exam asking the student to "use the repeated addition strategy to solve: 5 x 3." The student answered "5+5+5" ...
JRN's user avatar
  • 10.8k
23 votes
5 answers
1k views

Students using ambiguous notation

I've noticed that many of my calculus students (all college students) will write, e.g., $1/3x$ to mean $(1/3)x$. This is an inherently ambiguous notation which I'd like them to avoid. Is simply ...
Avi Steiner's user avatar
21 votes
5 answers
4k views

Grating mathematical phrases---How to correct?

As mathematics educators, we all have come across students using mathematical notation incorrectly (looking at you, $\frac{d}{dx}$ vs $\frac{dy}{dx}$ or $\frac{\infty^2}{\infty}$). My question focuses ...
erfink's user avatar
  • 1,129
21 votes
5 answers
788 views

How students write their work, and learning outcomes

While teaching students mathematics, I have noticed that some seem sloppy in the way that they write down their work. For example, a student is given a question: What is the area of the rectangle? <...
ctrl-alt-delor's user avatar
20 votes
3 answers
3k views

Why are $m$ and $b$ used in the slope-intercept equation of a line?

The slope-intercept form of the equation of a line is often presented in textbooks (in the US) as $$y = mx + b\,,$$ where $m$ is the slope of the line and $b$ is the $y$-intercept. How did $m$ and $...
Mike Pierce's user avatar
  • 4,845
20 votes
5 answers
1k views

Why is multiplication taught using cross notation at first?

Alert: I am not a math educator. It seems to me that multiplication is first taught using the cross notation, for example $3\times 5=15$. First question - is that even correct? Maybe not all schools ...
Džuris's user avatar
  • 757
20 votes
7 answers
1k views

Notation Conflict between Teachers and Textbooks

In mathematics notation plays an important role in clarifying the subject. A bad notation could be confusing. Recently I use a logic textbook which has a very nice approach and content but an ...
user avatar
17 votes
4 answers
828 views

Do students confuse $\log_ab$ and $\log a^b$?

I recently observed a group of students being introduced to logarithms for the first time. Some of them had trouble writing $\log_ab$ properly, and it looked more like $\log a^b$. All logarithms have ...
Joonas Ilmavirta's user avatar
15 votes
4 answers
2k views

Why do we use functional composition in the order we do?

Function composition means, roughly, taking the output of a function and applying it to the input of another function. If we define an object C to represent this operation, we could say $C(f,g) = f∘g$ ...
David Lalo's user avatar
15 votes
7 answers
587 views

How to teach brackets?

I was taught in a school that one has to use different brackets in expressions like $\{[(3+4)\cdot 4]^4\}^{1/2}$ to denote the order which subexpression is evaluated first. But can this be recommended ...
beginnertutor's user avatar
14 votes
4 answers
5k views

Parentheses around negative numbers

We teach students that a notation like $$17 - -59$$ is not acceptable or at least not good. Instead we want them to write $$17-(-59)$$ The main reason seems to be that it's more readable if you ...
realGottlob's user avatar
14 votes
6 answers
640 views

Is using different notations in one course a good idea?

From aesthetical point of view, using two symbols for the same concept during the same course is obviously a very bad idea. However, especially when I teach freshmen, I often deliberately mix ...
mbork's user avatar
  • 1,299
14 votes
2 answers
2k views

The use of "$\therefore$" and "$\because$"

In schools, many students learn the usage of "$\therefore$" and "$\because$" in proofs. Such three-dot notation are popular in many high-school books and exams, but are almost ...
Ma Joad's user avatar
  • 1,673
14 votes
6 answers
4k views

Explaining the symbols in definite and indefinite integrals

I teach the definite integral before the indefinite for a few reasons, one of which is that I want students to recognize that the definite integral means area (not anti-derivative). If we do ...
Sue VanHattum's user avatar
  • 20.8k
14 votes
4 answers
1k views

As a TA, how to reduce imprecise notations/statements in students' exams

I'm not a course instructor, just a TA of the first quarter calculus course who lead discussion sections and grade exams. When grading the midterm, I found large number of students showed some ...
user avatar
13 votes
4 answers
637 views

Using $dx$ for $h$ in the definition of derivative

Is it mathematically correct to write $$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx},$$ rather than $$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}?$$ If not, what is the difference? If so, why isn't this ...
John Molokach's user avatar
13 votes
5 answers
1k views

Writing Fractions "Correctly"

I very often see students writing, for example, $1/3x$ when they mean $\frac 13x$. I used to tell them not to write $1/3x$ beause it looks like $\frac{1}{3x}$ until I realized that, according to ...
Steven Alexis Gregory's user avatar
12 votes
5 answers
2k views

A different symbol for the indefinite integral/antiderivative?

Examples. An indefinite integral (or antiderivative) of $\cos$ is $\sin$: $$\int \cos = \sin.$$ Edit: There has been much unexpected confusion with the above statement. I define the above statement ...
user avatar
12 votes
1 answer
390 views

Why do some of my, usually international/Indian students, write limits to the left of the integral?

I see a lot of my students (I am in the US), usually Indian, write the limits of integration to the LEFT of the integral sign rather than customary top or right. The formula will look for example like ...
user avatar
11 votes
8 answers
13k views

Proof of why BODMAS (or BIDMAS) works?

In my first full-time teaching post, it is very likely that I'll need to be teaching a small amount of GCSE Mathematics to students retaking it. One thing that has been bugging me is that I can't seem ...
user avatar
11 votes
6 answers
2k views

What is the right notation to use in multivariable chain rules?

The following "chain rule" is in my multivariable calculus course: If $f$ depends on $x$ and $y$, but $x$ and $y$ depend on $t$, then $\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{d ...
Chris Cunningham's user avatar
11 votes
4 answers
1k views

Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ -- why?

I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than ...
user avatar
11 votes
3 answers
530 views

Are there any negative consequences in applying operations/functions to a whole equality?

Some of my students solve equations not by applying the same operations on the left and right sides of an equation, but by applying the operation to the whole equality. For example, they may write ...
Taladris's user avatar
  • 1,491
11 votes
4 answers
4k views

Notation for change of basis matrix

As far as I can tell, it's only a slight exaggeration to say that every text has a different notation for a change of basis matrix from (say) $\mathcal{B}$ to $\mathcal{C}$. That's not even to talk ...
kcrisman's user avatar
  • 5,976
11 votes
1 answer
1k views

How to denote angle?

I'm teaching mathematics on my free time for young pupils. Once I have seen that they denote angles like $\angle ABC$. But sometimes I have difficulties to understand whether they mean an angle or its ...
freetimetutor's user avatar
11 votes
1 answer
12k views

When and Why are different division symbols taught?

There are 4 division symbols that I have learned/taught. Below is 18 divided by 3, shown with 4 different symbols. This question was sparked by the comments on my answer to the question on examples ...
Amy B's user avatar
  • 8,017
11 votes
3 answers
640 views

Terminology for parts of limit notation

When we talk about: $$\lim_{x\to{c}}f(x)=L.$$ Is there a formal name for the number "$c$"? I know that the notation means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It ...
Ari's user avatar
  • 379
10 votes
5 answers
2k views

Lowercase vs. uppercase letters for matrix entries

For a matrix $A$ in, say for instance, $\mathbb{R}^{m \times n}$, there are at least two different conventions to denote its entry at position $(j,k)$: Denote the entry as $a_{jk}$. Denote the entry ...
Jochen Glueck's user avatar
10 votes
1 answer
696 views

Language to Distinguish Between Variables and Arbitrary Constants

Today in second semester calculus, I found myself stumbling a bit to provide a natural-sounding explanation for all the letters involved in the expression $$ \lim_{t \rightarrow \infty} \int_1^t \frac{...
Austin Mohr's user avatar
10 votes
2 answers
5k views

Notation of points with coordinates

At least in Germany, nearly all teachers and textbooks use the notation $$P(x,y)$$ for the point $P$ with coordinates $x$ and $y$. My own math professors at university always cried about this, as the ...
Toscho's user avatar
  • 3,342
10 votes
2 answers
714 views

Is this example of Leibniz notation sloppy?

In helping a family member who is studying calculus, I was asked about the meaning of the following, which is straight out of a calculus text book (Varberg and Purcell) $v'(t) = \frac{{\rm d}\,v}{{\...
Theodore Norvell's user avatar
9 votes
2 answers
2k views

Explaining Sigma-Notation

I attempted to introduce the summation notation $\Sigma$ to my students. The notation was unfamiliar to the students beforehand. I worked through many examples with them, but for most of them, working ...
wayne's user avatar
  • 193
9 votes
7 answers
2k views

Is there a resource for learning to read mathematical notation/equations/formulae?

Ideally, I am looking for an online resource. But a book or any other would help already. Background: I am a senior teaching assistant in the field of business and statistics. Most of my students have ...
Nibood_the_slightly_advanced's user avatar
9 votes
3 answers
412 views

Framework for Compound Inequalities

I have been presenting compound inequalities like $3 < x < 7$ as being a shorter way of saying $3 < x$ and $x < 7$. From this point of view, though, I end up having to admit that ...
Chris Cunningham's user avatar
9 votes
3 answers
1k views

Is there a standard convention for interpreting ambiguous absolute value expressions?

Consider the expression $$|x + 2|x + 3|x + 4|.$$ One way to interpret this is that there are two products being added together: $$|x+2|x \hspace{1cm} + \hspace{1cm} 3|x+4|$$ But you could also ...
Justin Skycak's user avatar
9 votes
6 answers
537 views

Helping high school students remember inequalities and division

Background I sometimes tutor high school students and I have come across various problem types that are best represented by the following two problems. They are unable to keep track of the correct ...
recmath's user avatar
  • 418
9 votes
2 answers
399 views

Question about function notation

In the textbook I am using to teach mathematics to high school students I found the following illustration about composition of functions. I do not agree with this illustration. For me $g$ is the ...
Ferenc Beleznay's user avatar
9 votes
2 answers
626 views

Fear of notation and hazily-appeared writing in Mathematics

I am looking for literature related to fear of notation in mathematics. It is even heard that the font size and font type make a reader reluctant to study mathematical literature, often lecture notes,...
Consider Non-Trivial Cases's user avatar
8 votes
5 answers
2k views

In teaching mathematics, should one always follow some international standards such as ISO 80000-2?

ISO 80000-2:2009 is a standard describing mathematical signs and symbols developed by the International Organization for Standardization (ISO). In teaching mathematics, should one always follow this ...
Zuriel's user avatar
  • 4,275