Questions tagged [notation]
For questions about good use of notation, comparison of specific notation, motivation of notation.
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $C(f,g) = f∘g$ ...
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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,...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 + \...
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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}...
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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}=...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{d ...
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Fear of notation and hazily-appeared writing in Mathematics
I am looking for literature related to fear of notation in mathematics.
It is even heard that the font size and font type make a reader reluctant to study mathematical literature, often lecture notes,...
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The use of "$\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 ...
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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 of course that it means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It just would be nice to be ...
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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\...
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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 ...
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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 ...
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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), ...
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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 ...
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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 ...
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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 $...
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Notation in the definition of matrix multiplication
When matrix multiplication is introduced, it is usually introduced with an additional variable: Given two multiplicable matrices $A$, $B$, one defines the product $C=AB$ to be the matrix given by some ...
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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 ...
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Are there standard notations for 'number talks' / ‘math talks?'
I’m a homeschool teacher of a nine-year-old, and we sometimes have one-on-one ‘number talks’ (a.k.a. 'math talks') similar to the activity used in primary school classrooms.
Part of this process ...
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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 ...
quid♦
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How should one approach the concept of "plus or minus", such as in the numerator of the quadratic formula?
The numerator is structured like:
$$(-b)\pm\sqrt{b^2- 4ac}.$$
Is it confusing or acceptable to distinguish between the following two things?
An idiom; and
What is or seems to be a compositionally ...
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Teaching asymptotic notations at the beginning of calculus [duplicate]
I'm thinking about teaching calculus by firstly introducing the asymptotic notations (big-Oh, little-oh, and $\sim$), secondly explaining their "arithmetic" (things like how to sum little-oh's and ...
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