Questions tagged [notation]
For questions about good use of notation, comparison of specific notation, motivation of notation.
51
questions
39
votes
18answers
5k 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 ...
2
votes
1answer
136 views
Naming arbitrary constants: subscripts, primes, or just more letters?
When choosing names for arbitrary constants either during a lesson or while working with a single student, should one use$\{n_1,n_2,n_3,\dotsc\}$ or $\{n, n', n'', \dotsc\}$ or $\{a,b,c,\dotsc\}$? Is ...
3
votes
1answer
115 views
What is the notation for polynomial long division in Norway?
I will be teaching a calculus-type course in Norwegian. Our textbook is unfortunately in English (the curse of a small language), but any custom exercises should be and all exams have to be in ...
5
votes
3answers
170 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 ...
12
votes
3answers
462 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 ...
6
votes
3answers
243 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 ...
1
vote
1answer
153 views
Can $y^{(n)}$ be used as a way of representing higher order derivatives?
I have never seen this notation, but I think that it follows in a similar vein for function notation. So if $y=f(x)$, then $y''=f''(x)$.
Then by that, can we say that
$$f^{(n)}(x)=y^{(n)}$$
18
votes
5answers
497 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?
<...
8
votes
1answer
202 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 ...
7
votes
2answers
505 views
Is “hat notation” for unit vectors commonly used in mathematics?
As an undergraduate, I clearly remember learning and using "hat notation" to describe unit vectors. That is, if $\vec{v}$ is any vector (in 2 or 3 dimensions) then $\hat{v}$ denotes the unit vector ...
16
votes
5answers
720 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 ...
13
votes
5answers
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 ...
14
votes
3answers
483 views
Explaining to students why $m$ and $b$ are used in the slope-intercept equation of a line
The slope-intercept form of the equation of a line is often presented in textbooks as
$$y = mx + b\,,$$
where $m$ is the slope of the line and $b$ is the $y$-intercept. How did $m$ and $b$ become ...
4
votes
2answers
132 views
What is the role of the efforts to change the fundamentals of maths? [closed]
One is often drawn to offbeat mathematical ideas and how they could revolutionize mathematics or at least make maths more easy to learn.
Current examples are:
Rational Trigonometry
Tau
Dozenal
...
6
votes
3answers
234 views
Wording VS mathematical notations
Is it better to write everything in words as the concepts themselves should be known? Or will some teachers in some countries prefer to be able to choose questions which also test the student's ...
18
votes
5answers
3k 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 ...
12
votes
1answer
327 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 ...
5
votes
2answers
135 views
At what educational stage are angles greater than 180 introduced?
Prompted by the question,
"How to denote angle?,"
I am interested to learn when students consider and reason with
angles $> 180^\circ$.
For example, when do they reason with an angle of $270^\...
10
votes
1answer
564 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 ...
7
votes
3answers
187 views
Design of a math exam using multiple choice or computer
I'm teaching math in first years of university (matrix algebra, differential analysis, etc.) since maybe 5 years, and I usually give written (paper) exams to the students.
I'm looking for ideas to ...
27
votes
6answers
4k 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 ...
6
votes
2answers
190 views
How to reasonably denote lines, line segments and rays?
I'm teaching geometry at high school for the first time soon and am struggling to find a reasonable notation for lines, line segments and rays defined by two points $A$, $B$ (and a direction). At the ...
4
votes
2answers
3k 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 ...
11
votes
8answers
8k 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 ...
3
votes
1answer
228 views
How to explain “fractional terms”?
as I can see there are mainly two ways to introduce fractional terms. Two examples to demonstrate the two variants:
$\frac{a^2+3}{a}; \frac{3}{2c}$
$T(a) = \frac{a^2+3}{a}; T(c) = \frac{3}{2c}$.
In ...
15
votes
4answers
480 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 ...
10
votes
1answer
2k 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 ...
1
vote
1answer
85 views
Moving lines in schematic diagrams
I am self-teaching myself precalculus.My biggest sticking point is to move lines within a diagram, especially, functions in the coordinated plane. For instance, doing horizontal and vertical shifts of ...
13
votes
4answers
870 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 ...
23
votes
7answers
4k 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" ...
12
votes
4answers
435 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 ...
12
votes
2answers
198 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 ...
7
votes
3answers
200 views
What symbol is appropriate to represent a sum modulo N? [closed]
I am writing a blog post about cryptography and I need to show some examples that involve sums modulo N.
I would like (if possible) to use a different "plus" symbol for these sums. The kind of ...
8
votes
2answers
201 views
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 / \...
5
votes
5answers
261 views
Notation for an element in a polynomial ring
Let $F$ be a field. What is the best notation (in an undergraduate or graduate abstract algebra class) for a generic element of the univariate polynomial ring $F[x]$?
The most common notation seems ...
11
votes
1answer
519 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
$$
\lim_{t \rightarrow \infty} \int_1^t \frac{...
8
votes
2answers
3k views
Notation of points with coordinates
At least in Germany, nearly all teachers and textbooks use the notation
$$P(x,y)$$
for the point $P$ with coordinates $x$ and $y$.
My own math professors at university always cried about this, as the ...
13
votes
6answers
2k 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 ...
8
votes
6answers
336 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 ...
29
votes
1answer
1k 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 ...
7
votes
1answer
118 views
Resource about notation for students
Others have already here pointed out that students can struggle with notation in mathematics. I can often think that the lack of proper notation gets in the way of solving a problem correctly.
...
9
votes
2answers
491 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}{{\...
4
votes
1answer
114 views
On using different notations for the same objects
Historically, in set theory we use two different notations to refer set theoretically same objects $\aleph_{\alpha}$ and $\omega_{\alpha}$. The folklore justification of this dual notation is that we ...
28
votes
6answers
1k 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 ...
1
vote
1answer
149 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 ...
13
votes
6answers
555 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 ...
22
votes
5answers
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 ...
19
votes
7answers
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 ...
3
votes
1answer
178 views
Better use “Integrate a function”/“Calculate the definite integral” or use terms like “primitive function”, “antiderivative”, “Aufleitung” (German)?
When one wants to let students calculate (Riemann) integrals in calculus, what is a good term to call this task?
If you want to focus on what the main task is, you may call it "Calculate the definite ...
33
votes
9answers
1k 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 ...
