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bio website tc.columbia.edu/academics/…
location Boston University, SED
age 28
visits member for 2 years, 4 months
seen 5 hours ago

Profile for Benjamin Dickman on Stack Exchange: MESE, MSE, and MO

Postdoctoral Fellow, Mathematics Education, Boston University SED.

Ph.D., Mathematics Education, Columbia University. 2014. NSFGRFP.

Dissertation: Conceptions of Creativity in Elementary School Mathematical Problem Posing.

M.Phil., Mathematics Education, Columbia University. 2014.

Fulbright Grant, Mathematics/Mathematics Education, Nanjing Normal University. 2008-2009.

Research Topic: High School Mathematics Teacher Training in China.

B.A., Mathematics, Amherst College. 2008. Honors Program.

Undergraduate Thesis: On a Theorem of Dwork. (p-adic proof of the rationality part of the Weil Conjectures.)


Have you solved any of my puzzles? If so, send me an email!
Email: bmd2118[at]colunbia.edu. (Change n to m.)


Aug
19
comment Most harmful heuristic?
@PedroTeixeira Surely this is the way to proceed! Let $y = x - 2.5$; then $x = y + 2.5$, and the equation becomes: $0 = (y + 1.5)(y + 0.5)(y - 0.5)(y - 1.5) = (y^2 - 2.25)(y^2 - 0.25)$, which holds when $y = \sqrt{2.25}$ or $y = \sqrt{0.25}$. For each square root we obtain two possible $y$-values; add back the $2.5$ to each to get the four possible $x$-values. A similar approach can be found by observing $(x-1)(x-4) = (x-2)(x-3) - 2$; now denote the LHS by $y$ so that the original equation becomes $y(y+2) = 0$; solve for $y$ using the quadratic equation, etc.
Aug
10
comment Examples of research on how people perceive mathematical objects
Does Jean Piaget's research (volume conservation tasks etc) exemplify what you are looking for here? What about Dor Abrahamson's work on proportions (youtube.com/watch?v=n9xVC76PlWc)? I'm not quite sure what you mean by mathematical objects, nor am I clear at what age/stage of mathematical development your focus lies. Probably a fair amount could be dug up from math education research on mathematical modeling, depending on how your question is interpreted...
Aug
8
comment Behaviour of power series on their circle of convergence
(Just a comment, assuming I haven't misunderstood.) To show that $\sum_{n} \frac{z^n}{n}$ converges for $C \ni z \neq 1$ (as you mention is shown in Rudin using trigonometric estimates) can be done very quickly with the Dirichlet Convergent Test, or -- even better! -- with the quick geometric proof provided here: mathoverflow.net/q/109582/22971
Jul
30
comment Examples of cubic Julia sets
@AdamEpstein Yes; I just recall this as the "most interesting" explicit cubic from the investigation. For an example of a cubic polynomial with points that don't escape, one could try $\phi(z) = \frac{1}{12}z^3 - \frac{25}{12}z + 1$, and check the forward orbit of the $5$-periodic point $z = 1$ (thereby making $0$ preperiodic). I don't think this is particularly helpful to the OP - though perhaps there is something of value in the paper - hence my leaving this as a mere comment.
Jul
29
comment Examples of cubic Julia sets
I participated in an REU that investigated cubic polynomials from the perspective of preperiodic points and canonical heights (the third section is on filled Julia sets). The arXiv link is arxiv.org/abs/0807.0468v3 and one interesting example is $f(z) = -\frac{25}{24}z^3 + \frac{97}{24}z + 1$; to see why, consider the forward orbit of $z = -\frac{7}{5}$...
Jul
18
answered Negative impact of wrong or non-rigorous proofs
Jul
12
comment What is known about tiling a rectangle in an irreducible way by smaller rectangles?
You might check Klarner's Theorem (Thm. 5) and its corollary in: Klarner, D. A. (1969). Packing a rectangle with congruent $N$-ominoes. *Journal of Combinatorial Theory, 7*(2), 107-115.
Jul
10
awarded  Enlightened
Jul
2
awarded  Curious
Jul
1
comment Is every set class generic over a given inner model?
I expect you would find the last section, beginning on p. 80, of math.berkeley.edu/~steel/papers/steel1.pdf to be helpful.
Jun
30
awarded  Nice Answer
Jun
30
revised Colourings of $\mathbb Q\times \mathbb Q$ in three colours
Corrected misspelling of 'Monsky' and made final remark slightly more precise.
Jun
30
answered Colourings of $\mathbb Q\times \mathbb Q$ in three colours
Jun
20
comment What's your favorite equation, formula, identity or inequality?
You might be interested in this answer: math.stackexchange.com/a/779515/37122
Jun
17
comment Review of Tim Maudlin's New Foundations for Physical Geometry
@NikWeaver I'm not a mathematical physicist, so I don't think my conclusions would serve much for the purpose of the OP. That said, the very first section gives an argument that $(0,1]$ is not open (in the sense of the standard topology on $\mathbb{R}$) with a foot-note attributing the argument to the Wikipedia page on Topology from 2005. That about summarizes my own personal feelings on the book preview...
Jun
16
comment Review of Tim Maudlin's New Foundations for Physical Geometry
You can read a fair amount of it on Google Books; probably enough to draw your own conclusions: books.google.com/books?id=10XbAgAAQBAJ
Jun
15
revised Is $x \, \tan(x)$ integrable in elementary functions?
Added in a full Liouville argument that int xtanx dx is not elementary
Jun
14
comment Enumeration of a finite group
@KevinO'Bryant The OP writes s/he hasn't found any example when $n$ is odd. When $n$ is even, consider $\mathbb{Z}/4\mathbb{Z}$ where $g_1 = 0, g_2 = 1, g_3 = 2, g_4 = 3$. Then $a_1 = 0, a_2 = 1, a_3 = 3, a_4 = 2$.
Jun
12
answered Frechet differentiable implies reflexive?
Jun
12
comment Origin of the term “generic” in set theory
@AsafKaragila The reason I remarked about Cohen's thesis is that his dissertation was entitled Topics in the Theory of Uniqueness of Trigonometric Series. See also the citation for a different piece on trigonometric series by Zygmund at the end of Cohen's On a Conjecture of Littlewood and Idempotent Measures. I think it's a pretty safe bet that he would have read the paper I mentioned above, but insignificant in the sense that a single foot-note with the word "generic" (used colloquially) is unlikely to have left a deep impression. (And I didn't root out other uses Cohen might have seen.)