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Results tagged with mathematics-education
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user 766
For questions in Mathematics Education as a scientific discipline. For more hands-on questions on teaching Mathematics, please use the tag teaching. There is also a Stack Exchange community http://matheducators.stackexchange.com/
15
votes
Insightful books about elementary mathematics
If first-order logic counts as "elementary mathematics", then I would like to suggest (the relevant chapters of) "Godel, Escher, Bach", by Douglas Hofstadter. (As an aside: Hofstadter's puzzle of enc …
94
votes
Accepted
Are there proofs that you feel you did not "understand" for a long time?
As an undergraduate, I learned the Sylow theorems in my algebra classes but could never retain either the statement or proof of these theorems in memory except for short periods of time (and in partic …
16
votes
Reference for a nice proof of "undetermined coefficients"
If $L$ has characteristic polynomial $\lambda \mapsto p(\lambda)$, then the conjugated differential operator $e^{-qt} L e^{qt}$ has characteristic polynomial $\lambda \mapsto p(\lambda+q)$. From thi …
- 114k
11
votes
Pedagogical question concerning $\Gamma(z)$
I guess it depends on whether the objective is (a) to introduce and motivate the Gamma function, and only the Gamma function, in as efficient a manner as possible, or (b) to present some useful mathem …
- 114k
141
votes
Accepted
Is the boundary $\partial S$ analogous to a derivative?
The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S_r|$ of the $r$-neighbourhoods $S_r$ of $S$ at $r=0$:
$$ |\partial S| = \frac{d}{dr} |S_r| |_{r=0}.$ …
- 114k
88
votes
Examples of common false beliefs in mathematics
This is perhaps a misunderstood definition rather than a false belief, but:
"A subnet of a net $( x_\alpha )_{\alpha \in A}$ takes the form
$( x_\alpha )_{\alpha \in B}$ for some subset $B$ of $A$. …
244
votes
Why is a topology made up of 'open' sets?
The textbook presentation of a topology as a collection of open sets is primarily an artefact of the preference for minimalism in the standard foundations of the basic structures of mathematics. Thi …
59
votes
What are your experiences of handouts in mathematics lectures?
Without pre-typed lecture notes (or a textbook that is being followed reasonably closely), many students often feel pressured to copy down every scrap that the lecturer writes down, in case they are m …
380
votes
Examples of common false beliefs in mathematics
The closure of the open ball of radius $r$ in a metric space, is the closed ball of radius $r$ in that metric space.
In a somewhat related spirit: the boundary of a subset of (say) Euclidean space ha …
103
votes
Examples of common false beliefs in mathematics
In order to show that a polynomial $P \in F[x_1,\ldots,x_n]$ vanishes, it suffices to show that $P(x_1,\ldots,x_n) = 0$ for all $x_1,\ldots,x_n \in F$. True in infinite fields, but very false for sma …
207
votes
Examples of common false beliefs in mathematics
Some false beliefs in linear algebra:
If two operators or matrices $A$, $B$ commute, then they are simultaneously diagonalisable.
(Of course, this overlooks the obvious necessary condition that each …
48
votes
Examples of common false beliefs in mathematics
The commutator [H,K] of two subgroups H,K is the set of commutators [h,k] with h in H and k in K. (Instead, it is the group generated by those commutators. Confusingly, the convention with products …
26
votes
How do you motivate a precise definition to a student without much proof experience?
I once asked my honours real analysis class to define the concept of an integer to a hypothetical bright young kid who was already perfectly familiar with the natural numbers and the operations one co …
21
votes
Real-world applications of mathematics, by arxiv subject area?
math.NA Numerical analysis
Linear programming algorithms are used in compressed sensing, which is now being used in MRI and imaging to increase resolution and/or decrease the number of measurements r …
6
votes
Real-world applications of mathematics, by arxiv subject area?
math.GM General Mathematics
The cosmic distance ladder is largely built using elementary geometry (although for some legs of the ladder, more advanced mathematics, e.g. relativity and probability, pl …