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

2
votes
1answer
135 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 $$f^{(n)}(x)=y^{(n)}$$
16
votes
5answers
296 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
183 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
304 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
644 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
432 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
126 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
180 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 ...
17
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
314 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
114 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
358 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
182 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 ...
24
votes
5answers
3k 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
177 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
2k 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 ...
10
votes
8answers
6k 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
214 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 ...
14
votes
4answers
420 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
1k 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
74 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
803 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 ...
22
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
422 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
181 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
176 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
192 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
237 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
499 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{...
7
votes
2answers
2k 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 ...
12
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 ...
7
votes
6answers
312 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 ...
26
votes
1answer
983 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
114 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
460 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
108 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 ...
27
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
144 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 ...
11
votes
6answers
483 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 ...
20
votes
4answers
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
168 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 ...
31
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 ...
27
votes
7answers
918 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 ...