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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/
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 …
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 …
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 $A$. …
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 …
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 …
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
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 …
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 …
30
votes
Real-world applications of mathematics, by arxiv subject area?
math.AG Algebraic geometry
Elliptic curve cryptography
Motion planning: Configuration spaces of robot arms are semi-algebraic sets, and algebraic geometry (especially the Cylindrical Algebraic Decomp …
31
votes
Real-world applications of mathematics, by arxiv subject area?
math.ST Statistics
almost everywhere
Accurate polling
Fraud detection: whether financial (e.g. via Benford's law) or voter fraud.
Used in the world of finance, economics and gambling on a daily basis …
46
votes
Real-world applications of mathematics, by arxiv subject area?
math.DG Differential geometry
Lie groups are used in robotics (to find the most efficient way to maneuver a robotic arm, for instance).
Spherical trigonometry is essential for navigation (a few centu …
17
votes
Real-world applications of mathematics, by arxiv subject area?
math.SP Spectral theory
Spectroscopy (of course)
To avoid bridge collapse, knowing where the resonant frequencies are is extremely important. :-)
Shape and model recognition
Network analysis and sec …
64
votes
Real-world applications of mathematics, by arxiv subject area?
math.AC Commutative algebra
Reed-Solomon codes (a type of error correction codes based on polynomials over finite fields - this is why CDs and DVDs still work even after being scratched!)