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Theron Hitchman
Works at University of Northern Iowa
Attended University of Michigan
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Theron Hitchman

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Theron Hitchman

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I am reading about differential equations these days, especially Cartan's theory of exterior differential systems. 

I stumbled upon a memorial piece about Cartan written by Shiing-Shen Chern and Claude Chevalley, written shortly after Cartan's death in 1951. One line about teaching jumped off the page at me.

Cartan was an excellent teacher; his lectures were gratifying intellectual experiences, which left the student with a generally mistaken idea that he had grasped all there was on the subject.

um...
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+Vincent Knight 'the illusion of learning is far worse than the absence of learning' -- I am totally stealing that. 

Also, this reminds me of this study: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3369499/ TLDR: Direct instruction significantly inhibited creative/exploratory behavior among preschoolers. 
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This was a pleasant thing to stumble across. 

This +Numberphile video features Dick Canary. I took a few classes from Dick when I was a graduate student at Michigan back in [mumble, mumble, mumble]. He was (and I assume still is) a great guy, and this "sports in hyperbolic space" was his typical intro. I remember the bit about golf lasting a lot longer, since he liked to golf. Anyway, enjoy!
 
We'll have more of this next week...

Patreon people usually get an earlier look... http://www.patreon.com/numberphile
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For the last year and a half, I have been working on a book, to be published by Johns Hopkins University Press. It will be a popular mathematics book, hopefully accessible to everyone. The twist is that whenever it makes sense, figures in the book will be photos of 3D prints that readers can download and print for themselves, buy online, or rotate around on screen.
The book will hopefully be out around September 2016. If you'd like a sneak peek, the work-in-progress website for the book (thanks +Jared Shay for helping me with it!) is at: http://3dprintmath.com
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How fun!
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Grading continues. Two classes down. One to go.

Deadline in 13 hours.
This is gonna be close.
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Yeah. Grades in by noon! Kinda amazing.
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Neat!
 
Besides making Rocket Sounds, this is what Astronauts do the whole day at the ISS.
Join the Simple Science and Interesting Things Community and share interesting stuff! https://plus.google.com/communities/117518490246975838002

The ISS serves as a microgravity and space environment research laboratory in which crew members conduct experiments in biology, human biology, physics, astronomy, meteorology and other fields. The station is suited for the testing of spacecraft systems and equipment required for missions to the Moon and Mars. The ISS maintains an orbit with an altitude of between 330 and 435 km (205 and 270 mi) by means of reboost manoeuvres using the engines of the Zvezda module or visiting spacecraft. It completes 15.54 orbits per day
http://en.wikipedia.org/wiki/International_Space_Station

http://img0.joyreactor.com/pics/post/fun-gif-zero-gravity-2099273.gif
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Physics would have been a lot easier to discover if humans grew up in zero gravity.
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Theron Hitchman

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Fancy View on a Simple Differential Equation

To fix ideas in my head, I made a picture which shows off the conceptual framework for solving differential equations with exterior differential systems for a very simple example: y' = x.

Of course, I used +The SageMathCloud to generate the images. Since I made this, I might as well share it.

Notes on what is going on:

Think of a differential equation as a condition F(x,y,y') = 0 for some function F. Since F is a function of three variables, we can plot this level set! The key is to use z = y'. Then F(x,y,z) is no big deal. Our task will be to find a curve on this level set which represents the solution function y=y(x).

But we don't want to forget that z is really the y' direction. How does this matter? Well, if z = y' = dy/dx, that means that the allowed slopes are in that z direction. It becomes a problem to think about where to put that in the picture. 

Following Cartan, the resolution is this:
z = dy/dx so...
z*dx - dy = 0.
Let w = z*dx - dy. This is a differential 1-form. At each point in space, it is a function that eats tangent vectors and spits out numbers. We
need a set-up where that is equal to zero... so at each point, we are only interested in the kernel of w. But that kernel is a plane! 
If we collect all of these planes together, we get something called a distribution. 

We can easily find a curve on the level set, but if we are going to do it in a way so that z = y', we must have that our curve is always tangent to the planes in this funny distribution.

To look closely, you can browse this SMC worksheet.
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Contact geometry FTW!
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Huh. That is freaky.

Via +Dave Pentecost
 
Albino redwoods are extremely rare, with only 61 known examples. Unable to produce chlorophyll, they live as parasites by fusing their roots with the roots of other redwood trees.
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I don't know if it should be called parasitism: http://goo.gl/4dVKhE
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Was reviewing a paper about something teaching related, so...

A Note to my Colleagues in Higher Education

the following things are different, you have do to them all, and it can help if you separate them in your head:
1. Formative Assessment
2. Feedback
3. Summative Assessment

Thank you, that is all.
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Time between release of grades and first question from a student about why is my grade such-and-such: less than an hour
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I had one student (I was a lab assistant in grad school and had grading duties) who asked me why he got 19/20 and not 20/20. Again, 14/20 contributes towards a first class honour for the course, and he got 19 for a scrap of paper with almost no context or explanation, no definitions, etc., and just a few rows of symbols with no relational symbols like implies or equals relating the rows. Still, the work was head and shoulders above the rest of the class. My answer was, "lack of perfection", and that seemed to end the debate.
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whoa! this is really interesting.

I should be grading. somebody +1 this so it shows up in my notifications later!
 
In this fascinating result, scientists resurrect an ancestral phenotype (dinosaur teeth) that has been lost in the species we know today (chicken). The result is a chicken with teeth, confirming that the hypothesized molecular mechanism for the major evolutionary transition from teeth to beak.

Background: Dinosaurs didn't totally disappear from earth. They live on today as birds. The reptiles you see today are a relatively separate line of evolution. We can group all birds and reptiles into a monophyletic group (thus birds really are reptiles), and the birds would be today's remnant of dinosaurs.

But birds have beaks. That means that some developmental pathways changed that used to be used for teeth development. To test this idea, scientists inhibited certain pathways thought to be responsible for this transition to beakiness, and the result was a chicken with teeth. The ancestral dinosaur phenotype has been restored.

So this is a fascinating result speaking to the mechanisms behind major evolutionary transitions.

Here is the primary source:

"A molecular mechanism for the origin of a key evolutionary innovation, the bird beak and palate, revealed by an integrative approach to major transitions in vertebrate history"
by Bhullar et al.
Evolution (2015), online first
http://dx.doi.org/10.1111/evo.12684

Abstract
=====
The avian beak is a key evolutionary innovation whose flexibility has permitted birds to diversify into a range of disparate ecological niches. We approached the problem of the mechanism behind this innovation using an approach bridging paleontology, comparative anatomy, and experimental developmental biology. First we used fossil and extant data to show the beak is distinctive in consisting of fused premaxillae that are geometrically distinct from those of ancestral archosaurs. To elucidate underlying developmental mechanisms, we examined candidate gene expression domains in the embryonic face: the earlier frontonasal ectodermal zone (FEZ) and the later midfacial Wnt-responsive region, in birds and several reptiles. This permitted the identification of an autapomorphic median gene expression region in Aves. In order to test the mechanism, we used inhibitors of both pathways to replicate in chicken the ancestral amniote expression. Altering the FEZ altered later Wnt responsiveness to the ancestral pattern. Skeletal phenotypes from both types of experiments had premaxillae that clustered geometrically with ancestral fossil forms instead of beaked birds. The palatal region was also altered to a more ancestral phenotype. This is consistent with the fossil record and with the tight functional association of avian premaxillae and palate in forming a kinetic beak.
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[ #evolution #science #biology #paleontology #dinosaurs #development #evodevo #devoevo #nature #research #news #birds #chicken #teeth #beak #developmentalbiology #evolutionarybiology ]
A chicken embryo with a dinosaur-like snout instead of a beak has been developed by scientists
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Thanks for the plusses. I just posted my grades, so now I can read.
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I asked students to construct a regular octagon as part of the final exam. I got one with an excellent choice of notation.
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Haha, awesome! That one's actually thinking.
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Work
Occupation
Professor of Mathematics
Employment
  • University of Northern Iowa
    Professor of Mathematics, 2007 - present
  • Rice University
    VIGRE Lovett Instructor of Mathematics, 2003 - 2006
  • Williams College
    Visiting Assistant Professor, 2006 - 2007
Basic Information
Gender
Male
Story
Tagline
College Mathematics Teaching, only mostly insane.
Introduction
I am interested in:
Math (especially geometry),
Inquiry Based Instruction,
Standards based assessment,
learning Python,
the Sage math software project,
Soccer,
good books.
Education
  • University of Michigan
    Mathematics, 1997 - 2003
  • THE Ohio State University
    Mathematics, 1993 - 1997
Contact Information
Work
Phone
+1 3192732646
Email
Address
Department of Mathematics 0506 University of Northern Iowa Cedar Falls, IA 50614-0506