Questions tagged [notation]
For questions about good use of notation, comparison of specific notation, motivation of notation.
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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}=...
user5108
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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 ...
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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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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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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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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 ...
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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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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 ...
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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 ...
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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" ...
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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 ...
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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 ...
user13395
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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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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?
<...
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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 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 ...
user230
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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 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 ...
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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 ...
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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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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 ...
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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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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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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 ...
user2139
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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 ...
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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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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 ...
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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 ...
user12641
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answer
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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 ...
user5108
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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 ...
user5447
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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 ...
user507
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3
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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 ...
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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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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 ...
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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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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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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
$$
\lim_{t \rightarrow \infty} \int_1^t \frac{...
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Is this example of Leibniz notation sloppy?
In helping a family member who is studying calculus, I was asked about the meaning of the following, which is straight out of a calculus text book (Varberg and Purcell)
$v'(t) = \frac{{\rm d}\,v}{{\...
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Notation of points with coordinates
At least in Germany, nearly all teachers and textbooks use the notation
$$P(x,y)$$
for the point $P$ with coordinates $x$ and $y$.
My own math professors at university always cried about this, as the ...
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When and Why are different division symbols taught?
There are 4 division symbols that I have learned/taught.
Below is 18 divided by 3, shown with 4 different symbols.
This question was sparked by the comments on my answer to the question on examples ...
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Framework for Compound Inequalities
I have been presenting compound inequalities like
$3 < x < 7$
as being a shorter way of saying
$3 < x$ and $x < 7$.
From this point of view, though, I end up having to admit that ...
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Helping high school students remember inequalities and division
Background
I sometimes tutor high school students and I have come across various problem types that are best represented by the following two problems.
They are unable to keep track of the correct ...
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Fear of notation and hazily-appeared writing in Mathematics
I am looking for literature related to fear of notation in mathematics.
It is even heard that the font size and font type make a reader reluctant to study mathematical literature, often lecture notes,...
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Terminology for parts of limit notation
When we talk about: $$\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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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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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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Is there a good notation for "ratio" comparable to the use of $\Delta$ for "difference"?
It is standard to use the symbol $\Delta$ to indicate a change in a quantity between two points on a curve, two rows on a table, and so forth. For linear functions, we write slope = $\Delta y / \...
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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 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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