81 votes
Accepted

Issues with "equals", where does this come from and how do I combat it?

A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or ...
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  • 4,755
49 votes

Proof of why BODMAS (or BIDMAS) works?

It's purely a matter of how we choose to define the notation. The main reason for it is that it lets us write polynomial expressions (which are extremely common) without parentheses, e.g., $x^3 + 3x^2 ...
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  • 4,755
46 votes

Should we stop differentiating between ln and log?

Unrelated to US, in Germany the notation through school and university was quite consistent: $\ln$ using base $e$ $\log$ using base $10$ $\log_x$ using base $x$ I may not know enough about the US ...
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33 votes
Accepted

Should students be asked to use more than one notation for the derivative in an introductory calculus class?

The different notations are to a considerable extent outgrowth of slightly different ways of looking at "the derivative." To me these different ways is the important thing to discuss, the notation is ...
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  • 7,572
30 votes

Misuse of parentheses for multiplication

It is actually wrong to say that parenthesis means multiplication. In $(2)(5)$ it is the lack of an operator between the parenthesis that implies multiplication, NOT the parenthesis. The parenthesis ...
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  • 401
29 votes
Accepted

Students using ambiguous notation

A colleague of mine includes on all his tests a line that reads (something like) "you will be graded on what you actually write down, not what I think you may have meant by what you wrote." He spends ...
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  • 1,102
27 votes
Accepted

Using $dx$ for $h$ in the definition of derivative

To use $$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx}$$ is mathematically correct if $dx$ is the name for a real variable. (If it should be something else it needs to be made clear what it should be.) ...
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  • 7,572
27 votes
Accepted

Grating mathematical phrases---How to correct?

Personally, I don't think we attend to this sufficiently in lower-level mathematics (where it's actually needed most). Students need that vocabulary to interface with books, future teachers, tutors, ...
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25 votes

Reasons for (not) distinguishing $f$ from $f(x)$

Consider the following observations: People shorten things with increasing frequency of usage. For example, the most frequent words are short. This is a way of making communication more effective, e....
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  • 8,729
24 votes

Reasons for (not) distinguishing $f$ from $f(x)$

It was mentioned in other answers that having a more sloppy notation is better to not complicate the communication. This is okay for people who had really understood the concepts of mathematics. I ...
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  • 9,096
24 votes
Accepted

Why are $m$ and $b$ used in the slope-intercept equation of a line?

This is also borderline not-an-answer, but it might be a nice broadening of your students' worldview to know that the "$m$" and the "$b$" are not universally accepted. Showing them this map (even ...
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  • 19.1k
24 votes
Accepted

Writing Fractions "Correctly"

I think that, depending on the maturity level of the students, you could just talk to them about why writing $\frac{1}{3}x$ rather than $1/3x$ makes it much clearer what you mean. They should ...
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  • 4,362
24 votes

How to help new students accept function notation

You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun ...
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  • 241
23 votes

How to help new students accept function notation

Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples: The function that maps every non-empty list to its first element. ...
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  • 570
22 votes

Reasons for (not) distinguishing $f$ from $f(x)$

It is interesting to see how computer algebra systems deal with this kind of thing. In Maple, for example, you can do the following: Define f := x^3 Enter ...
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22 votes

How to help new students accept function notation

You should tell them these two main benefits: (1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable ...
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  • 496
22 votes

Should we stop differentiating between ln and log?

You are wrong about undergraduate courses always treating $\log$ as $\ln$. To my memory, all of my undergrad chemistry and physics (not just general, but majors texts), engineering, calculus, diffyQs, ...
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  • 237
21 votes
Accepted

Metonymy in mathematics

Metonymy and its relatives, metaphor, polysemy, synecdoche occur all over the place in mathematical writing, and sometimes cause students problems and sometimes don't, because those thought processes ...
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21 votes
Accepted

Allowing nonstandard mathematical language and/or notation

I think, while teaching, the principal way to judge mathematical language is not whether it's standard, but whether it's effective communication. This difference applies principally to communication ...
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20 votes
Accepted

Reasons for (not) distinguishing $f$ from $f(x)$

To distinguish not too strictly between $f$ and $f(x)$ allows to operate more easily with functions built up from other functions. For example, one might want to say things like: Let us consider ...
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  • 7,572
20 votes
Accepted

Misuse of parentheses for multiplication

To answer the ultimate question ("Can anybody explain where this writing tradition comes from?"): It's explicitly taught that way by many U.S. instructors and textbooks. Examples: From the otherwise ...
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20 votes

Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ -- why?

When a student writes incorrect notation, ask them to read it out loud. I would say something like: Something here doesn't look right, but we can fix it. Could you read this work out loud? I think ...
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  • 19.1k
20 votes

Should we stop differentiating between ln and log?

I would argue that we should never use $\log$ for $\log_{10}$ anymore, only warn that this was historically often done. Sticking to the ISO convention is probably safest: $$\begin{aligned} \log_{e} ...
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19 votes

Should we stop differentiating between ln and log?

In mathematics, $\log$ means natural logarithm. So, as you become a mathematician, sometime during that process you must learn this. That standards document is for natural sciences, not for ...
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  • 6,171
19 votes

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

Common knowledge The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these ...
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  • 4,544
19 votes
Accepted

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

I think the basic answer is that there are all sorts of things that we could write, but don't. We usually leave off things that are redundant, but we can add them back in when convenient. For ...
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  • 919
18 votes

Misuse of parentheses for multiplication

I disagree that it is "terribly harmful". Do not prevent them from writing $(2)(5)$. Instead prevent them from writing things that are actually wrong. Thinking that $\sin x$ is $\sin$ times $x$ ...
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  • 6,171
17 votes
Accepted

Framework for Compound Inequalities

Just because you've defined a meaning for $a < b < c$ does not mean that any mishmash of other relational operators becomes equally well-defined as notation. Stick with what you've defined for a ...
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17 votes

What is the right notation to use in multivariable chain rules?

What's wrong with this?: $$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$ with $\frac{dt}{dt}...
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  • 4,656

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