This is a guest post by Courtney Gibbons, an assistant professor of mathematics at Hamilton College. You can see her teaching evaluations on ratemyprofessor.com. She would like you to note that she’s been tagged as “hilarious.” Twice.
Lately, my social media has been blowing up with stories about gender bias in higher ed, especially course evaluations. As a 30-something, female math professor, I’m personally invested in this kind of issue. So I’m gratified when I read about well-designed studies that highlight the “vagina tax” in teaching (I didn’t coin this phrase, but I wish I had).
These kinds of studies bring the conversation about bias to the table in a way that academics can understand. We can geek out on experimental design, the fact that the research is peer-reviewed and therefore passes some basic legitimacy tests.
Indeed, the conversation finally moves out of the realm of folklore, where we have “known” for some time that students expect women to be nurturing in addition to managing the class, while men just need to keep class on track.
Let me reiterate: as a young woman in academia, I want deans and chairs and presidents to take these observed phenomena seriously when evaluating their professors. I want to talk to my colleagues and my students about these issues. Eventually, I’d like to “fix” them, or at least game them to my advantage. (Just kidding. I’d rather fix them.)
However, let me speak as a mathematician for a minute here: bad interpretations of data don’t advance the cause. There’s beautiful link-bait out there that justifies its conclusions on the flimsy “hey, look at this chart” understanding of big data. Benjamin M. Schmidt created a really beautiful tool to visualize data he scraped from the website ratemyprofessor.com through a process that he sketches on his blog. The best criticisms and caveats come from Schmidt himself.
What I want to examine is the response to the tool, both in the media and among my colleagues. USAToday, HuffPo, and other sites have linked to it, citing it as yet more evidence to support the folklore: students see men as “geniuses” and women as “bossy.” It looks like they found some screenshots (or took a few) and decided to interpret them as provocatively as possible. After playing with the tool for a few minutes, which wasn’t even hard enough to qualify as sleuthing, I came to a very different conclusion.
If you look at the ratings for “genius” and then break them down further to look at positive and negative reviews separately, it occurs predominantly in negative reviews. I found a few specific reviews, and they read, “you have to be a genius to pass” or along those lines.
[Don’t take my word for it — search google for:
rate my professors “you have to be a genius”‘
and you’ll see how students use the word “genius” in reviews of professors. The first page of hits is pretty much all men.]
Here’s the breakdown for “genius”:
Similar results occur with “brilliant”:
Now check out “bossy” and negative reviews:
I thought that the phrase “terrible teacher” was more illuminating, because it’s more likely in reference to the subject of the review, and we’ve got some meaningful occurrences:
Who’s doing this reporting, and why aren’t we reading these reports more critically? Journalists, get your shit together and report data responsibly. Academics, be a little more skeptical of stories that simply post screenshots of a chart coupled with inciting prose from conclusions drawn, badly, from hastily scanned data.
Is this tool useless? No. Is it fun to futz around with? Yes.
Is it being reported and understood well? Resounding no!
I think even our students would agree with me: that’s just f*cked up.
Aunt Pythia has something in the works for you dear people, but it’s not quite ready yet, and you’ll have to wait another week. Rest assured, it will be worth it. And apologies to mathbabe.org subscribers who received an errant test post this week.
In the meantime, Aunt Pythia is going to write a quick column today from a Montreal hotel room after an amazing workshop yesterday which she will comment on later in the week.
So quick, get some tea and some flannel-lined flannel, because damn it’s wintery outside, all snowy and shit. Aunt Pythia’s about to spew her usual unreasonable nonsense!
LET’S DO THIS PEOPLES!!! And please, even if you’ve got nothing interesting to say for yourself, feel free to make something up or get inspired by Google auto complete and then go ahead and:
ask Aunt Pythia your question at the bottom of the page!
Dear Aunt Pythia,
This may not really be an “Aunt Pythia” question. But could either you or Mathbabe comment on this article on sexism in academic science?
I can imagine many ways they could be misrepresenting the statistics, but I don’t know which.
No Bias, Really?
Dear No Bias,
I was also struck by the inflammatory tone and questionable conclusions of this article. But you know, controversy sells.
So, here are a couple of lines I’ll pull out. First:
Our country desperately needs more talented people in these fields; recruiting more women could address this issue. But the unwelcoming image of the sexist academy isn’t helping. Fortunately, as we have found in a thorough analysis of recent data on women in the academic workplace, it isn’t accurate, either.
Many of the common, negative depictions of the plight of academic women are based on experiences of older women and data from before the 2000s, and often before the 1990s. That’s not to say that mistreatment doesn’t still occur — but when it does, it is largely anecdotal, or else overgeneralized from small studies.
I guess right off the bat I’d ask, how are you collecting data? The data I have personally about sexist treatment at the hands of my colleagues hasn’t, to my knowledge, been put in any database. The sexist treatment I’ve witnessed for pretty much all of my female mathematics colleagues has, equally, never been installed in a database to my knowledge. So yeah, not convinced these people know what they are talking about. It’s famously hard to prove something doesn’t exist, especially when you don’t have a search algorithm.
One possibility for the data they seem to have: they interviewed people after the fact, perhaps decades after the fact. If that’s the case, then you’d expect more and better data on older women, and that’s what we are currently seeing. There is a lag on this data collection, in other words. That’s not the same as “it doesn’t exist.” A common mistake researchers make. They take the data as “objective truth” and forget that it’s a human process to collect it (or not collect it!). Think police shootings.
The article then goes on to talk about how the data for women in math and other science fields isn’t so bad in terms of retention, promotion, and other issues. For there I’d say, the women have already gone through a mighty selection process, so in general you’d expect them to be smarter than their colleagues, so in general their promotion rates should be higher, but they aren’t. So that’s also a sign of sexism.
I mean, whatever. That’s not actually what I claim is true, so much as another interpretation of this data. My overall point is that, they have some data, and they are making strong and somewhat outrageous claims which I can dismiss without much work.
I hope that helps!
Dear Aunt Pythia,
In his November “Launchings” column, David Bressoud has presents some interesting data on differences between male and female college calculus students. As much as I’ve appreciated all of Bressoud’s careful explorations of mathematics education, I find I’m a bit irritated by his title, “MAA Calculus Study: Women Are Different,” because it appears to take the male experience as the norm.
Perhaps I was already annoyed because of a NYTimes op-ed, “Academic Science Isn’t Sexist”, in which Wendy Williams and Steven Ceci claim that “[w]e are not your father’s academy anymore,” and that the underrepresentation of women in math-intensive fields is “rooted in women’s earlier educational choices, and in women’s occupational and lifestyle preferences.” Here, too, the message seems to be “don’t worry about changing the academy — women are different from the norm, which is (naturally) that which works for men.”
My question for you, Aunt Pythia, is this: am I overreacting here?
I received my PhD in mathematics in 1984, and I’ve seen significant change for the better in the academy since then. Child care at AMS meetings? A crowd in the women’s rest room at same? Unthinkable when I started. But if women are still disproportionately “choosing” to go into other fields, might we look a little more closely at the environments in which girls and women are making their educational and “lifestyle” choices?
I welcome your thoughts. If you’re eager for more data analysis, I’d also love to hear your take on the paper by Williams, Ceci, and their colleagues.
Still One of the Underrepresented After All These Years
Without even reading that article, I can say without hesitation that yes, it’s a ridiculous title, and it’s infuriating and YOU ARE NOT OVERREACTING. To be clear, that is bold-faced, italicized, and all caps. I mean it.
The word “different” forces us to compare something to a baseline, and given that there is no baseline even mentioned, we are forced to guess at it, and that imposes the “man as default” mindset. Fuck that. I mean, if the title had been, “There are differences between male and female calculus students,” I would not have been annoyed, because even though “male” comes first, I’m not a stickler. I just want to acknowledge that if we mention one category, we mention the other as well.
To illustrate this a bit more, we don’t entitle a blog post “Whites are different” and leave it at that, because we’d be like, different from whom? From blacks? From Asians? From Asian-Americans? See how that works? You need to say different from some assumed baseline, and the assumed baseline has to be a cultural norm. And right now it’s white male. Which is arguable one reason that calculus students act differently when they are men (har!).
As for the other article, I already shit on that in the previous answer but I’m happy to do it once again. It’s bullshit, and I’m disappointed that the Times published it.
As for the article, I don’t have time now but I’ll take a look, thanks!
Dear Aunt Pythia,
I am twenty years old, near the halfway point in my senior year of a mathematics BS at a large, well-regarded public university in the Northeast. I’ve been aiming my energies at graduate school, and I am now looking at PhD program applications. Most apps ask for two or three letters of recommendation from a faculty member who is familiar with your work. This poses a very big problem, because all of my professors hate me.
Okay, maybe it’s not quite like that. But I’ve had a really lousy time in the math department at LWRPUN. My fellow students are dispassionate, unresponsive, and unfriendly. My professors are dry, uncommitted to their students, and the ones who aren’t mathematically incompetent are lousy teachers. On top of all this, a crippling bureaucracy has prevented me countless times from taking classes I’m interested in (few as they are in this catalog), substituting instead ANOTHER REQUIRED SEMESTER OF ANALYSIS.
So I haven’t made any personal connections of the sort that might benefit me in the form of a letter of rec. My work hasn’t even been that good; my depression and anxiety (in general as well as re all this) have increasingly prevented me from completing even easy homework assignments. Nobody here has seen my best mathematical work, and for that matter, nobody anywhere else has either*.
And for four years, everyone I’ve come to with this gathering creeping progressively life-eating concern has given me the same old BS about You should really put yourself out there! and It’s just so important to go to your professor’s office hours! without considering maybe — I’ve tried, I really have.
What can I do, Aunt Pythia? I’m really passionate about mathematics, but I’m worried I won’t be able to pursue my studies without these magic papers.
Reports Embargoed by Crummy Lecturers, Earnestly Seeking Solace
*I thankfully have a professor from an outside experience willing to write about my teaching credentials, but that one letter is surely not sufficient to show my potential as a graduate student and researcher.
I am afraid I will have to call bullshit on you, RECLESS. Plus your sign-off doesn’t actually spell anything.
Here’s the thing, there are no mathematically incompetent lecturers at large, well-regarded public universities. There are, in fact, mathematically very competent people who can’t get jobs at such places. Such is the pyramid-shaped job market of mathematics. So whereas I believe you when you say your lecturers have been uninspired, and uncommitted to their students, the fact that you added “mathematically incompetent” just makes me not believe you at all, in anything.
Here’s what I think is happening. You think you’re really into math, but you’ve never really understood your classes, nor have you understood that you’ve never understood your classes, because your self-image is that you’re already a mathematician, and that people have just not acknowledged your brilliance.
But that’s not how math actually works. Math is a social endeavor, where you have to communicate your ideas well enough for others to understand them, or else you aren’t doing math.
I’m not saying you haven’t had bad luck with teachers. It’s a real possibility. But there’s something else going on as well, and I don’t think you can honestly expect to go to the next level without sorting stuff out. In other words, even if you don’t love the teacher, if you loved the subject, got into it, and did the proofs, you’d still be getting adequate grades to ask for letters. The thing about writing letters, as a math prof, is that you don’t have to like the student personally to write a good letter, you just need to admire their skills. But since you can’t do that either, you won’t get good letters, and moreover I don’t think you’d deserve good letters. And therefore I don’t think you should go to grad school.
Suggestion: look carefully at your own behavior, figure out what it is you are doing that isn’t working. Maybe think of what you love about math, or about your own image of being a mathematician, and see if there’s something you really know you’re good at, and other people know it to, and develop that.
Dearest Aunt Pythia,
I have a sex question for you! Kind of. You have to get through the boring back story first…I’m a 19 year old female physics major. I’m quiet, rather mousy, and awkward. A lot of the time I feel like I have more to prove than the boys do, because I’m a girl, and because of the aforementioned shyness.
People seem to automatically assume I’m unintelligent. I think I’m just as intelligent as the boys in my program, but I don’t come off that way! Point is, I want to be this cool, strong, independent, successful, respectable girl who doesn’t take shit from misogynistic people who assume I’m inferior.
However, I feel extremely guilty about my sexual preferences. I’m pretty submissive. I’d like power exchange in my relationships…hair pulling, bondage, spanking, being bossed around, the whole bit. I like to be dominated by men. Older men. Smart older men. Hopefully I’ve successfully conveyed my dilemma. I want to be respected by the men (and women, and others) I’m surrounded by in my academic life, but taken control of as a girlfriend.
Why does what I despise happening to me in an academic setting please me so much in a romantic/sexual one? Agh, I feel like such a bad girl! (and not in the arousing way…)
This is such a relief – finally, a sex question! – and it’s honestly one of the best questions I’ve ever gotten, ever, in Aunt Pythia or elsewhere. I’m so glad I can answer this for you.
It is absolutely not in conflict to want something in a sexual context that is abhorrent to you in normal life. It is in fact a well-known pattern! You shouldn’t feel at all weird about it! Lots – LOTS – of the submissives I’ve met are, in their day jobs, the boss, literally. They have companies and are extremely fancy and in control. And then they love to be bossed around and spanked. Seriously. If anything, my feeling is that your sexual proclivities point to being alpha in real life, but maybe I’m going overboard.
So yeah, no problem here. You are killing it. And in 3 or 4 years I want you to write back and explain to me how you’ve found an amazing lover who gives you what you want in the bedroom and worships your physics prowess outside it. There will, in fact, be people lining up for this role.
And those people in your program? Do your best to ignore them. Men are just impossibly arrogant at that age, but time will humble them somewhat even as your confidence will rise as you learn more. I’m not saying it ever evens out entirely but it does improve.
Also: find other women (and super cool men) to study with. Surround yourself with supportive people. Take note of obnoxious people and avoid them. Trade up with friends whenever possible.
Well, you’ve wasted yet another Saturday morning with Aunt Pythia! I hope you’re satisfied! Please if you could, ask me a question. And don’t forget to make an amazing sign-off, they make me very very happy.
Click here for a form or just do it now:
Today I want to discuss a recent review in New York Review of Books, on a new book entitled Who’s Afraid of the Big Bad Dragon? Why China Has the Best (and Worst) Education System in the World by Yong Zhao (hat tip Alex). The review was written by Diane Ravitch, an outspoken critic of No Child Left Behind, Race To The Top, and the Common Core.
You should read the review, it’s well written and convincing, at least to me. I’ve been studying these issues and devoted a large chunk of my book to the feedback loops described as they’ve played out in this country. Here are the steps I see, which are largely reflected in Ravitch’s review:
- Politicians get outraged about a growing “achievement gap” (whereby richer or whiter students get better test scores than poorer or browner students) and/or a “lack of international competitiveness” (whereby students in countries like China get higher international standardized test scores than U.S. students).
- The current president decides to “get tough on education,” which translates into new technology and way more standardized tests.
- The underlying message is that teachers and students and possibly parents are lazy and need to be “held accountable” to improve test scores. The even deeper assumption is that test scores are the way to measure quality of learning.
- Once there’s lots of attention being given to test scores, lots of things start happening in response (the “feedback loop”).
- For example, widespread cheating by students and teachers and principals, especially when teachers and principals get paid based on test performance.
- Also, well-off students get more and better test prep, so the achievement gap gets wider.
- Even just the test scores themselves lead to segregation by class: parents who can afford it move to towns with “better schools,” measured by test scores.
- International competitiveness doesn’t improve. But we’ve actually never been highly ranked since we started measuring this.
What Zhao’s book adds to this is how much worse it all is in China. Especially the cheating. My favorite excerpt from the book:
Teachers guess possible [test] items, companies sell answers and wireless cheating devices to students, and students engage in all sorts of elaborate cheating. In 2013, a riot broke out because a group of students in Hubei Province were stopped from executing the cheating scheme their parents purchased to ease their college entrance exam.
Ravitch adds after that that ‘an angry mob of two thousand people smashed cars and chanted, “We want fairness. There is no fairness if you do not let us cheat.”’
To be sure, the stakes in China are way higher. Test scores are incredibly important and allow people to have certain careers. But according to Zhao, this selection process, which is quite old, has stifled creativity in the Chinese educational system (so, in other words, test scores are the wrong way to measure learning, in part because of the feedback loop). He blames the obsession with test scores on the fact that no Chinese native has received a Nobel Prize since 1949, for example: the winners of that selection process are not naturally creative.
Furthermore, Zhao claims, the Chinese educational system stifles individuality and forces conformity. It is an authoritarian tool.
In that light, I guess we should be proud that our international scores are lower than China’s; maybe it is evidence that we’re doing something right.
I know that, as a parent, I am sensitive to these issues. I want my kids to have discipline in some ways, but I don’t want them to learn to submit themselves to an arbitrary system for no good reason. I like the fact that they question why they should do things like go to bed on time, and exercise regularly, and keep their rooms cleanish, and I encourage their questions, even while I know I’m kind of ruining their chances at happily working in a giant corporation and being a conformist drone.
This parenting style of mine, which I believe is pretty widespread, seems reasonable to me because, at least in my experience, I’ve gotten further by being smart and clever than by being exactly what other people have wanted me to be. And I’m glad I live in a society that rewards quirkiness and individuality.
I’ve been sent this recent New York Times article by a few people (thanks!). It’s called Grading Teachers, With Data From Class, and it’s about how standardized tests are showing themselves to be inadequate to evaluate teachers, so a Silicon Valley-backed education startup called Panorama is stepping into the mix with a data collection process focused on student evaluations.
Putting aside for now how much this is a play for collecting information about the students themselves, I have a few words to say about the signal which one gets from student evaluations. It’s noisy.
So, for example, I was a calculus teacher at Barnard, teaching students from all over the Columbia University community (so, not just women). I taught the same class two semesters in a row: first in Fall, then in Spring.
Here’s something I noticed. The students in the Fall were young (mostly first semester frosh), eager, smart, and hard-working. They loved me and gave me high marks on all categories, except of course for the few students who just hated math, who would typically give themselves away by saying “I hate math and this class is no different.”
The students in the Spring were older, less eager, probably just as smart, but less hard-working. They didn’t like me or the class. In particular, they didn’t like how I expected them to work hard and challenge themselves. The evaluations came back consistently less excited, with many more people who hated math.
I figured out that many of the students had avoided this class and were taking it for a requirement, didn’t want to be there, and it showed. And the result was that, although my teaching didn’t change remarkably between the two semesters, my evaluations changed considerably.
Was there some way I could have gotten better evaluations from that second group? Absolutely. I could have made the class easier. That class wanted calculus to be cookie-cutter, and didn’t particularly care about the underlying concepts and didn’t want to challenge themselves. The first class, by contrast, had loved those things.
My conclusion is that, once we add “get good student evaluations” to the mix of requirements for our country’s teachers, we are asking for them to conform to their students’ wishes, which aren’t always good. Many of the students in this country don’t like doing homework (in fact most!). Only some of them like to be challenged to think outside their comfort zone. We think teachers should do those things, but by asking them to get good student evaluations we might be preventing them from doing those things. A bad feedback loop would result.
I’m not saying teachers shouldn’t look at student evaluations; far from it, I always did and I found them useful and illuminating, but the data was very noisy. I’d love to see teachers be allowed to see these evaluations without there being punitive consequences.
I was was having a wonderful ramen lunch with the mathbabe and, as is all too common when two broad minded Ph.D.’s in math get together, we started talking about the horrible state math education is in for both advanced high school students and undergraduates.
One amusing thing we discovered pretty quickly is that we had independently come up with the same (radical) solution to at least part of the problem: throw out the traditional sequence which goes through first and second year calculus and replace it with a unified probability, statistics, calculus course where the calculus component was only for the smoothest of functions and moreover the applications of calculus are only to statistics and probability. Not only is everything much more practical and easier to motivate in such a course, students would hopefully learn a skill that is essential nowadays: how to separate out statistically good information from the large amount of statistical crap that is out there.
Of course, the downside is that the (interesting) subtleties that come from the proofs, the study of non-smooth functions and for that matter all the other stuff interesting to prospective physicists like DiffEQ’s would have to be reserved for different courses. (We also were in agreement that Gonick’s beyond wonderful“Cartoon Guide To Statistics” should be required reading for all the students in these courses, but I digress…)
The real point of this blog post is based on what happened next: but first you have to know I’m more or less one generation older than the mathbabe. This meant I was both able and willing to preface my next point with the words: “You know when I was young, in one way students were much better off because…” Now it is well known that using this phrase to preface a discussion often poisons the discussion but occasionally, as I hope in this case, some practices from days gone by ago can if brought back, help solve some of today’s educational problems.
By the way, and apropos of nothing, there is a cure for people prone to too frequent use of this phrase: go quickly to YouTube and repeatedly make them watch Monty Python’s Four Yorkshireman until cured:
Anyway, the point I made was that I am a member of the last generation of students who had to use slide rules. Another good reference is: here. Both these references are great and I recommend them. (The latter being more technical.) For those who have never heard of them, in a nutshell, a slide rule is an analog device that uses logarithms under the hood to do (sufficiently accurate in most cases) approximate multiplication, division, roots etc.
The key point is that using a slide rule requires the user to keep track of the “order of magnitude” of the answers— because slide rules only give you four or so significant digits. This meant students of my generation when taking science and math courses were continuously exposed to order of magnitude calculations and you just couldn’t escape from having to make order of magnitude calculations all the time—students nowadays, not so much. Calculators have made skill at doing order of magnitude calculations (or Fermi calculations as they are often lovingly called) an add-on rather than a base line skill and that is a really bad thing. (Actually my belief that bringing back slide rules would be a good thing goes back a ways: when that when I was a Program Director at the NSF in the 90’s, I actually tried to get someone to submit a proposal which would have been called “On the use of a hand held analog device to improve science and math education!” Didn’t have much luck.)
Anyway, if you want to try a slide rule out, alas, good vintage slide rules have become collectible and so expensive— because baby boomers like me are buying the ones we couldn’t afford when we were in high school – but the nice thing is there are lots of sites like this one which show you how to make your own.
Finally, while I don’t think they will ever be as much fun as using a slide rule, you could still allow calculators in classrooms.
Why? Because it would be trivial to have a mode in the TI calculator or the Casio calculator that all high school students seem to use, called “significant digits only.” With the right kind of problems this mode would require students to do order of magnitude calculations because they would never be able to enter trailing or leading zeroes and we could easily stick them with problems having a lot of them!
But calculators really bug me in classrooms and, so I can’t resist pointing out one last flaw in their omnipresence: it makes students believe in the possibility of ridiculously high precision results in the real world. After all, nothing they are likely to encounter in their work (and certainly not in their lives) will ever need (or even have) 14 digits of accuracy and, more to the point, when you see a high precision result in the real world, it is likely to be totally bogus when examined under the hood.
Everyone I know who codes uses stackoverflow.com for absolutely everything.
Just yesterday I met a cool coding chick who was learning python and pandas (of course!) with the assistance of stackoverflow. It is exactly what you need to get stuff working, and it’s better than having a friend to ask, even a highly knowledgable friend, because your friend might be busy or might not know the answer, or even if your friend knew the answer her answer isn’t cut-and-paste-able.
If you are someone who has never used stackoverflow for help, then let me explain how it works. Say you want to know how to load a JSON file into python but you don’t want to write a script for that because you’re pretty sure someone already has. You just search for “import json into python” and you get results with vote counts:
Also, every math nerd I know uses and contributes to mathoverflow.net. It’s not just for math facts and questions, either, there are interesting discussions going on there all the time. Here’s an example of a comment in response to understanding the philosophy behind the claimed proof of the ABC Conjecture:
OK well hold on tight because now there’s a new online forum, but not about coding and not about math. It’s about all the other STEM subjects, which since we’ve removed math might need to be called STE subjects, which is not catchy.
So far only statistics is open, but other stuff is coming very soon. Specifically it covers, or soon will cover, the following fields:
- Cognitive Sciences
- Computer Sciences
- Earth and Planetary Sciences
- Science & Math Education
- History of Science and Mathematics
- Applied Mathematics, and
I’m super excited for this site, it has serious potential to make peoples’ lives better. I wish it had a category for Data Sciences, and for Data Journalism, because I’d probably be more involved in those categories than most of the above, but then again most data science-y questions could be inserted into one of the above. I’ll try to be patient on this one.
Here’s a screen shot of an existing Stats question on the site:
Yesterday was a day filled with secrets and codes. In the morning, at The Platform, we had guest speaker Columbia history professor Matthew Connelly, who came and talked to us about his work with declassified documents. Two big and slightly depressing take-aways for me were the following:
- As records have become digitized, it has gotten easy for people to get rid of archival records in large quantities. Just press delete.
- As records have become digitized, it has become easy to trace the access of records, and in particular the leaks. Connelly explained that, to some extent, Obama’s harsh approach to leakers and whistleblowers might be explained as simply “letting the system work.” Yet another way that technology informs the way we approach human interactions.
After class we had section, in which we discussed the Computer Science classes some of the students are taking next semester (there’s a list here) and then I talked to them about prime numbers and the RSA crypto system.
I got really into it and wrote up an iPython Notebook which could be better but is pretty good, I think, and works out one example completely, encoding and decoding the message “hello”.