51
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 ...
- 4,843
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 ...
- 629
38
votes
Correcting how a student writes symbols
Personally, if I can make up an ordinary math problem where the student's alternate/new/strange symbols lead to an incorrect response, then I think that's grounds for correcting the student. (Of ...
- 1,291
36
votes
Parentheses around negative numbers
It is not a good idea to against a convention, unless you have strong reasons for it.
That said, you might not want to punish those going against the convention, either.
An obvious reason for having ...
- 6,110
32
votes
Accepted
Explaining Sigma-Notation
I've experienced positive results by first having students spend some time writing out sums in full (or using ellipsis notation if there are many terms).
That way, it gets annoying to spend so much ...
- 1,291
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 ...
- 401
28
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, ...
- 23.6k
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.)
...
quid♦
- 7,632
26
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 ...
- 1,169
25
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 ...
- 20.8k
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 ...
- 4,506
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 ...
- 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.
...
- 580
22
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 ...
- 23.6k
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 ...
- 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, ...
- 237
21
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} ...
- 693
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 ...
- 777
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 ...
- 20.8k
20
votes
Why do we still teach the determinant formula for cross product? And is it as bad as I think it is?
My answer is probably not very useful when teaching in high school. I'll just mention here a few reasons why this definition is in fact a good one, and why it's a good idea to teach this formula at a ...
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 ...
- 7,273
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 ...
- 6,110
19
votes
Accepted
Correcting how a student writes symbols
Not sure if this is the case but: is this student a Spanish speaker? What they write looks like ñ, so if that's the case it could just be that they are interpreting 𝜋 as a letter they know, specially ...
- 306
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$ ...
- 7,273
18
votes
Correcting how a student writes symbols
This isn't a moral conundrum, and your students shouldn't be snowflakes who freak out when they're corrected on something like this. They need your guidance in fixing their incorrect habits while they'...
- 197
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 ...
- 23.6k
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}$$
with $\frac{dt}{dt}...
- 4,915
17
votes
Accepted
How to teach using brackets in sums?
I would say that in terms of order of operations, the summation symbol is between multiplication/division and addition/subtraction. So when you write $\sum_i a_i+b_i$, the implication is to do $\sum_i ...
- 7,252
17
votes
Parentheses around negative numbers
Others have talked about legibility, so I want to address the importance of the parentheses in an expression like
$$17\color{red}{-}(\color{blue}{-}59)$$
from the perspective of educational psychology....
- 2,394
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