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Questions tagged [notation]

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

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4 votes
2 answers
282 views

Correct notation of a Sample space

From the very beginning I have used the notation $ S = \{ H , T \} $ as the Sample space for tossing a coin once and $ S = \{ HH , HT , TH , TT \} $ in the case of tossing a coin twice.I have several ...
7 votes
3 answers
366 views

Why is it written $\tan^{-1}$?

I was showing my neighbour's daughter how to do simple sine, cosine and tangent questions and their inverses. She could do the questions fine but she asked me why the calculator button for inverse tan ...
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 ...
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 ...
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 ...
-5 votes
1 answer
107 views

Mnemonics for memorizing the difference between a closed interval and an open interval

Here are some suggestions for memorizing the difference between an open interval and a closed interval. [a,b]={x|a<=x<=b} (a,b)={x|a<x<b} The first thing I would notice is that the bracket ...
8 votes
5 answers
404 views

Should one include the unit in the variable? E.g. should one write $x^\circ = 30^\circ$ or $x = 30^\circ$?

I sometimes come across equations like $x^\circ = 180^\circ - \frac{360^\circ}{n}$ where I would write $x = 180^\circ - \frac{360^\circ}{n}$, or like $3 \text{ cm} + x \text{ cm} = 5 \text{ cm}$ where ...
-4 votes
2 answers
271 views

Order of operations pemdas

Why was the order of operations established in mathematics with multiplication taking precedence over addition, as dictated by the PEMDAS rule? What historical or practical factors influenced this ...
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 ...
7 votes
4 answers
444 views

The word "numeral", is it being taught and does the word exist for it in your language?

I am a mathematics educator from Lithuania and I have recently realized that there is no separate word in our language for the word "numeral". To be more precise there is no term to describe the ...
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 ...
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 ...
3 votes
3 answers
257 views

Is there a preferred way to format a negative exponent?

Say there's an exam question whose answer is $x$ to the power of negative one. Two ways of writing this are $x^{-1}$ and $\frac{1}{x}$. I know that questions will sometimes request an answer without ...
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 ...
4 votes
0 answers
118 views

Activities that encourage students to create or evaluate mathematical notations

I'm looking for references about activities that encourage elementary school students to create or evaluate mathematical notations. do you know any?
7 votes
4 answers
4k views

What implication arrows, if any, should I require in teaching?

Q: Solve $x+5=0$. A: $x+5=0\implies x=-5$. This answer would be given full marks. Isn’t it better to tell students to use $\equiv$ or $\iff$? Because that is what lets them say $-5$ is a solution to ...
3 votes
5 answers
894 views

Simple examples of how a good notation or diagram can help to solve math problems

I am looking for simple examples of situations where a good notation/diagram was fundamental for solving an elementary problem in mathematics (I am looking for examples accessible to a basic school ...
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 ...
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 ...
2 votes
1 answer
408 views

Do Greek students use Greek letters to denote angles?

In western schools is a tradition to use Greek letters to denote angles. I wonder what about Greek schools do they also use Greek letters to denote angles or do they prefer other kind of alphabet to ...
3 votes
3 answers
1k views

How to teach using brackets in sums?

How one should teach using brackets in summation? For example, why is it correct to write $\sum_i a_ib_i$ but $\sum_i a_i+b_i$ should be written as $\sum_i (a_i+b_i)$ But $\sum_i\frac{a_i+b_i}{2}$ is, ...
2 votes
2 answers
235 views

How to introduce the use of Greek letters in high school?

I am looking for any hints or experience reports or materials/potential difficulties about how to introduce the use of Greek letters in high school Math/Physics.
0 votes
1 answer
241 views

Which are the most used Greek letters in math textbooks?

I am looking for a list of the most frequent Greek letters used in high school and college textbooks or some other corpora. I've realized my students don't know Greek letters and I would like to teach ...
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 ...
6 votes
3 answers
9k views

Notation of line segment and its length

I have sometimes seen a notation where $AB$ could mean either the line segment or its length. Why do the same notation can be mean both? Should one teach pupils to use for example notation $d(A,B)$ or ...
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 ...
4 votes
0 answers
125 views

What should I do when I, as a math tutor, use incorrect verbal notation?

I do freelance tutoring; the math-related ones are mostly 1-2 hour one-on-one sessions for high school and college students taking calculus classes. Often I find myself scrambling for words - I don't ...
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 ...
8 votes
0 answers
186 views

References for mathematical notation for foreign students in the U.S

I teach quite a few foreign students at a U.S. university. Frequently students are placed in our most remedial math class due to not having placement scores and failing to test out of the course. I ...
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 $C(f,g) = f∘g$ ...
1 vote
1 answer
324 views

Limit from both sides or from left? [closed]

Is it possible to write a problem statement as follows: A function $f$ is defined on $]0,1[$ as $f(x)=x$. Determine $\lim_{x\to 1}f(x)$. Or should one write always as: A function $f$ is defined on $]0,...
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 ...
6 votes
2 answers
361 views

Effective Strategies for Helping Students Recognize Nonsensical Expressions? (HS - Undergrad Level)

I'm not entirely how best to pose this question, so that it fits within the guidelines (so edits / suggestions for modification are warmly welcome). I'm interested in exploring effective strategies ...
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 ...
5 votes
3 answers
371 views

What's a good notation to show elements of relation composition?

Teaching discrete mathematics, we pose (from the textbook) questions on finding compositions of relations, notably, relations on very small finite sets with only 3 or 4 elements (as an introductory ...
7 votes
6 answers
2k views

Why do we write $x$ instead of $1x$?

I am currently student teaching for an Integrated Math 1 class (which is similar to Algebra 1) that consists of 9th graders. I have been teaching my students how to solve linear systems using ...
7 votes
7 answers
4k views

What is the preferred way to denote the Pythagorean theorem equation?

I am teaching 12-16 year olds. How should I write down the Pythagorean theorem equation? Some alternatives: $a^2 + b^2 = c^2$ $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$ $\text{leg}_1^2 + \...
4 votes
7 answers
349 views

Should I avoid writing: $ 11:40 - 15 \text{ min} = 11:25$, and what are alternatives to this way of writing?

I want to stress to my students that we should be careful with how we treat the equals sign and that we should always make sure that the units match. However, sometimes I write $ 11:40 - 15 \text{ min}...
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}=...
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 ...
2 votes
1 answer
365 views

Is there a conventional function notation that takes a polynomial and order and yields the coefficient corresponding to the order?

I am writing a book and for the sake of simplicity I want to do something as follows. Coef((-3x^2 +5x -1)(x^2 +1), 2) = -3 -1 = -4 where the first argument ...
6 votes
4 answers
755 views

Who actually uses $\mathbf i$, $\mathbf j$, $\mathbf k$ for the standard unit vectors?

I am wondering which research communities use the notation $\mathbf i$, $\mathbf j$, $\mathbf k$ for the three-dimensional unit vectors. The calculus textbook I have to use (Stewart) uses that ...
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 ...
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 ...
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,...
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 ...
1 vote
1 answer
240 views

What to call a symbol that denotes an "undisclosed" given number? [closed]

Students like to categorize notations to pin down their understanding of exactly what these notations stand for. Thus, given the expressions $f(x_{0})=f(x)|_{x\leftarrow x_{0}}$, $x=x_0+h$, or $lim_{x\...
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 ...
4 votes
3 answers
440 views

Students writing $f(x^2+1)$ when they probably mean $f(x)=x^2+1$

Over the past years teaching freshmen calculus I've repeatedly seen students make the following type of error: Suppose they have to express some quantity $y$ as function of $x$, when the relation ...
5 votes
2 answers
308 views

Undergraduate Vector Calculus Notation Mess

Question 1: What are your arguments in favor of the big array of different notations used in the context of undergraduate vector calculus: line integrals, surface integrals (of scalars and fields), ...