22
votes
Psychological test for Euclidean geometry
The FCI is a "concept inventory" test for classical mechanics. There exist several such tests for mathematics. Some may be implemented as "peer instruction" (as Eric Mazur ...
- 178k
11
votes
Psychological test for Euclidean geometry
This question is slightly easier than your questions but seems thematically of the sort you are aiming for.
Pat walks around the edges of a triangular park. After Pat has walked two edges, they go to ...
- 6,100
10
votes
Psychological test for Euclidean geometry
Since no kindergarten level answers have yet been proposed, I'll start a post with just one:
Standing in the middle a large flat field with a straight fence along one side, say to the child Run to ...
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8
votes
Examples of common false beliefs in mathematics
I just learned incidentally in the comments of another question here that it is not true that every proper subgroup is contained in a maximal (proper) subgroup. A counterexample is easy to find: the ...
6
votes
Examples of common false beliefs in mathematics
Let $A$ be an $n\times n$-matrix over a noncommutative (but associative and unital) ring. Then, $A$ is invertible if and only if its transpose $A^T$ is invertible, right?
Wrong. A counterexample (...
5
votes
Solving interval problems without outer measure
I give an elementary solution to Problem 1 in $\mathbb{R}^n$ in my book Measure Theory and Functional Analysis (Proposition 2.16, p. 48). Here is the one-dimensional version.
I guess it's clear that ...
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4
votes
The interrelationship problem of modern mathematics – How to deal with it in first year graduate courses?
This is an old question, but I only just stumbled across it.
I agree that it's important to convey to graduate students the inter-connectedness of mathematics. But I don't think that the way to do it ...
4
votes
Amount of mathematical knowledge required for starting Ph.D. in pure mathematics
American students are generally not ready to start PhD because of mediocre undergraduate education. This is not really a complain about the US education (which I think is great), but because most of ...
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3
votes
Psychological test for Euclidean geometry
This blog post points out psychometric issues with the Calculus Concept Inventory:
Spencer Bagley, Jim Gleason, Lisa Rice, Matt Thomas, Diana White, Does the Calculus Concept Inventory really measure ...
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3
votes
Video lectures of mathematics courses available online for free
I have recorded lecture series on
Random Matrices
Mathematical Aspects of Quantum Mechanics
High Dimensional Analysis: Random Matrices and Machine Learning
3
votes
Accepted
Book on analysis and algebra at the undergraduate level
I can recommend two books.
The first one is Simmon's 'Introduction to topology and modern analysis'.
I myself studied some portions of this book when an undergraduate.It has three parts. Part one is a ...
3
votes
PhD dissertations that solve an established open problem
Robin Moser in his PhD thesis found a constructive proof for the Lovasz Local Lemma, a problem that was essentially open for decades. This earned him a Godel Prize.
3
votes
Examples of common false beliefs in mathematics
I’m not sure if this counts as “reasonably advanced”, but given the wide-ranging nature of the answers here, I feel like this is worth a mention, given that it plagued my mind for so long when I was ...
2
votes
One-step problems in geometry
A the boundary of a convex polyhedron is shellable.
2
votes
What are your favorite instructional counterexamples?
For a function $f:\mathbb R^n\to\mathbb R^m$, it is possible for the directional derivatives in all directions to exist at a point without $f$ being continuous there, let alone differentiable. Let $f:\...
2
votes
Examples of common false beliefs in mathematics
A dense subspace of a Hilbert space $H$ must contain an ONB for $H$.
This is, of course, true if $H$ is separable, but false in general. See, for example, this answer in MSE: https://math....
2
votes
An example of a beautiful proof that would be accessible at the high school level?
I would put on such a list the proof of the Cauchy-Schwarz inequality directly from the axioms of an inner product using the discriminant criterion for the classification of the roots of a second-...
2
votes
Video lectures of mathematics courses available online for free
A series of lectures on symmetric functions, Macdonald polynomials and double affine Hecke algebras (videos and notes) organised in 2021 by R Venkatesh of the Indian Institute of Science; see also its ...
2
votes
Solving interval problems without outer measure
Concerning problem 2. I do not know whether this "elementary", but at least formally it does not use the notion of outer measure or Lebesgue measure.
Replace the closed intervals to open ...
- 103k
1
vote
How does a Masters student of math learn physics by self?
In case you are interested in general relativity, I suggest taking a look at a book I wrote, A Mathematical Introduction to General Relativity.
It presents general relativity to undergraduate ...
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1
vote
Solving interval problems without outer measure
Further to Nik Weaver's answer, which proves the case of $I$ bounded in Problem 1, the unbounded case is proved below - it follows as a corollary of the bounded case. I will update this answer later ...
1
vote
Short papers for undergraduate course on reading scholarly math
Conway and Gordon's paper Knots and Links in Spatial Graphs proves two interesting elementary theorems:
Any (non-pathological) embedding of the graph $K_6$ in 3D space contains two linked loops.
Any ...
1
vote
An example of a beautiful proof that would be accessible at the high school level?
I would also like to add a different proof of Fermats little theorem ($p|(a^p-a)$ for prime $p$) to the list.
Suppose you have a colors of pearls and you want to produce pearl-chains of length $p$.
...
1
vote
Examples of common false beliefs in mathematics
Root(s) of a $3^{rd}$ degree polynomial over $
\Bbb Q$ are expressible
using radicals with the imaginary $i$. If a root $r$ is real,
by taking only the real part, $r$ is expressible using radicals ...
1
vote
Accepted
One-step problems in geometry
Let $A$ be a set of intially labelled points in $\mathbb{R}^d$. We may take any line containing at least $k$ labelled points and label any point on this line. For which minimal size $|A|$ (as a ...
1
vote
One-step problems in geometry
I learned this one from my advisor: the Borromean rings are not realized by round circles.
Really this is two steps, since one needs to know that the Borromean rings are nontrivial (not the unlink), ...
1
vote
One-step problems in geometry
Von Neumann's law for two-dimensional cell growth
A soap froth or polycrystalline slab is modeled by a two-dimensional network of piecewise smooth curves, joined at vertices at internal angles of $2\...
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