Inspired by Gregory Beyrer's post about equity and his "Summer of Canvas" plus it being the Fourth of July holiday, I am re-posting below an blog post from another blog: 10 Ways to Give Your Students the Gift of Slack. I've changed the title (a lot of people thought I meant Slack-the-app), and I've updated it with some links to Canvas Community spaces in which some of these same ideas have come up. I hope this is something that will promote more discussion and more blog posts; it's my opinion that designing-for-equity is both a pedagogical and a civic duty, and it is not just about technology or about online courses: it is about the future of public education in this country.
The cartoon that came along with this post was particularly poignant.
Your post reminded me of a challenge I see every time Couros posts about students having those three aspects of a digital identity: no matter how much we as educators may encourage this, ultimately it is up to the students to make it part of their lives. I have been blogging with my students for some years now, and when it is not a class requirement, they stop posting. I think part of this digital presence that we want students to establish – the \”residency,\” as Robert Schuetz said in the recent blog post that led me here (http://www.rtschuetz.net/2016/02/mapping-our-pangea.html) – is not always happening where we suggest. I know my students have an online presence – but it\’s on Instagram and Snapchat, not the blogsphere. Perhaps instead of dragging kids on vacation to where we think they should set up shop, we need to start following them to their preferred residences and help them turn those into sturdy, worthy places from which to venture out into the world.
This is certainly an intriguing way to look at it, but there’s another way to frame it as well. Students are on sites like Instagram and Snapchat because they’re connecting with their friends there. I doubt many (any?) are using those platforms for learning or engagement purposes, so attempting to engage with them there may not translate for educators. It may have the colloquial effect of “I’m on Snapchat because my parents aren’t; if my parents join I’m either going to block them or move to another platform they’re not on.” Something similar to this was seen in cultural teen use of Facebook as parents swarmed to the platform over the past decade. To slightly reframe it, how many high school teachers in the past have seen students in the hallways between classes socializing and thought to themselves, “I should go out and teach in the hallway, because that’s where the students are and they seem alert?”
It might also shed some light on our perspectives to look at what happens at the end of a quarter or semester in most colleges. I always remember book sellback time and a large proportion of my friends and colleagues rushed to the bookstore to sell their textbooks back. (I’ll stipulate the book market has changed drastically in the past two decades since I was in University, but I think the cultural effect is still roughly equivalent.) As a bibliophile I could never bring myself to sell books back because I felt the books were a significant part of what I learned and I always kept them in my personal collection to refer back to later. Some friends I knew would keep occasional textbooks for their particular area of concentration knowing that they might refer back to them in later parts of their study. But generalizing to the whole, most students dumped their notes, notebooks, and even textbooks that they felt no longer had value to them. I highly suspect that something similar is happening to students who are “forced” to keep online presences for coursework. They look at it as a temporary online notebook which is disposable when the class is over and probably even more so if it’s a course they didn’t feel will greatly impact their future coursework.
I personally find a huge amount of value in using my personal website as an ongoing commonplace book and refer back to it regularly as I collect more information and reshape my thoughts and perspectives on what I’ve read and learned over the years. Importantly, I have a lot of content that isn’t shared publicly on it as well. For me it’s become a daily tool for thinking and collecting as well as for searching. I suspect that this is also how Aaron is using his site as well. My use of it has also reached a fever pitch with my discovery of IndieWeb philosophies and technologies which greatly modify and extend how I’m now able to use my site compared to the thousands of others. I can do almost all of the things I could do on Facebook, Twitter, etc. including interacting with them directly and this makes it hugely more valuable to me.
The other difference is that I use my personal site for almost everything including a wide variety of topics I’m working on. Most students are introduced to having (read: forced to maintain) a site for a single class. This means they can throw it all overboard once that single class is over. What happens if or when they’re induced to use such a thing in all of their classes? Perhaps this may be when the proverbial quarter drops? Eventually by using such a tool(s) they’ll figure out a way to make it actively add the value they’re seeking. This kernel may be part of the value of having a site as a living portfolio upon graduation.
Another issue I often see, because I follow the space, is that many educational technologists see some value in these systems, but more often than not, they’re not self-dogfooding them the same way they expect their students to. While there are a few shining examples, generally many teachers and professors aren’t using their personal sites as personal learning networks, communications platforms, or even as social networks. Why should students be making the leap if their mentors and teachers aren’t? I can only name a small handful of active academic researchers who are heavily active in writing and very effectively sharing material online (and who aren’t directly in the edtech space). Many of them are succeeding in spite of the poor quality of their tools. Rarely does a day go by that I don’t think about one or more interesting thought leaders who I wish had even a modicum of online space much less a website that goes beyond the basic functionality of a broken business card. I’ve even offered to build for free some incredibly rich functional websites for researchers I’d love to follow more closely, but they just don’t see the value themselves.
I won’t presume to speak for Aaron, but he’s certainly become part of my PLN in part because he posts such a rich panoply of content on a topic in which I’m interested, but also in larger part because his website supports webmentions which allows us a much easier and richer method of communicating back and forth on nearly opposite sides of the Earth. I suspect that I may be one of the very few who extracts even a fraction of the true value of what he publishes through a panoply of means. I might liken it to the value of a highly hand-crafted trade journal from a decade or more ago as he’s actively following, reading, and interacting with a variety of people in a space in which I’m very interested. I find I don’t have to work nearly as hard at it all because he’s actively filtering through and uncovering the best of the best already. Who is the equivalent beacon for our students? Where are those people?
So the real question is how can we help direct students to similar types of resources for topics they’re personally interested in discovering more about? It may not be in their introduction to poetry class that they feel like it’s a pain doing daily posts about on a blog in which they’re not invested. (In fact it sounds to me just like the the online equivalent of a student being forced to write a 500 word essay in their lined composition book from the 1950’s.) But it’ll be on some topic, somewhere, and this is where the spark meets the fuel and the oxygen. But the missing part of the equation is often a panoply of missing technological features that impact the culture of learning. I personally think the webmention protocol is a major linkage that could help ease some of the burden, but then there’s also issues like identity, privacy, and all the other cultural baggage that needs to make the jump to online as seamlessly (or not) as it happens in the real world.
…perhaps we’re all looking for the online equivalent of being able to meld something like Maslow’s Hierarchy of Needs with Bloom’s Taxonomy?
I’ll have to expand upon it later, but perhaps we’re all looking for the online equivalent of being able to meld something like Maslow’s Hierarchy of Needs with Bloom’s Taxonomy? It’s certainly a major simplification, but it feels like the current state of the art is allowing us to put the lower levels of Bloom’s Taxonomy in an online setting (and we’re not even able to sell that part well to students), but we’re missing both its upper echelons as well as almost all of Maslow’s piece of the picture.
With all this said, I’ll leave you all with a stunningly beautiful example of synthesis and creation from a Ph.D. student in mathematics I came across the other day on Instagram and the associated version she wrote about on her personal website. How could we bottle this to have our students analyzing, synthesizing, and then creating this way?
Twenty states suspend people’s professional or driver’s licenses if they fall behind on loan payments, according to records obtained by The New York Times.
This has to be one of the most un-ethical and painfully stupid laws out there. Far better would be for them to focus their efforts at shutting down the predatory for-profit schools which are causing students to have some of these unpayable loans in the first place.
It’s almost as a nation like we’re systematically trying to destroy ourselves and our competitive stance within the world just for spite.
Following months of hard work, we are finally ready to publish our 2017 e-book, Education and Technology: critical approaches. This bilingual collection brings together 12 chapters written by researchers based in Brazil, Australia, Scotland, England and USA. The work has been edited by Giselle Ferreira, Alexandre Rosado e Jaciara Carvalho, members of the ICT in Educational Processes Research Group, who maintain this blog (mostly in Portuguese – at least so far!).
From the editors’ Introduction:
"This volume offers a measure of sobriety in reaction to the excesses and hyperboles found in the mainstream literature on Education and Technology. The pieces (…) tackle questions of power and consider contextual and historical specificities, escaping the usual euphoria that surrounds digital technology and adopting different perspectives on our current historical moment."
There are two obvious sources of funding and PR for “personalized learning” – the Gates Foundation and the Chan Zuckerberg Initiative. The former has spent hundreds of millions of dollars on “personalized learning” products and projects; the latter promises it will spend billions.
There are some out-sized influences in the education space. If only the US Government were better at pushing influence in this area…
Within the past week, two well-known and well-established coding bootcamps have announced they’ll be closing their doors: Dev Bootcamp, owned by Kaplan Inc., and The Iron Yard, owned by the Apollo Education Group (parent company of the University of Phoenix). Two closures might not make a trend… yet. But some industry observers have suggested we might see more “consolidation” in the coming months.
Some great observations on non-profit vs. for-profit educational institutions and the social inequality that exists between the two.
The books introduce subjects like rocket science, quantum physics and general relativity — with bright colors, simple shapes and thick board pages perfect for teething toddlers. The books make up the Baby University series — and each one begins with the same sentence and picture — This is a ball — and then expands on the titular concept.
In particular, some asked about alternate projects for basing education projects around which aren’t WordPress. Some suggested using WithKnown which is spectacular for its interaction model and flexibility. I suspect that many in the conversation haven’t heard of or added webmentions (for cross-site/cross-platform conversations or notifications) or micropub to their WordPress (or other) sites to add those pieces of functionality that Known comes with out of the box.
Another section of the conversation mentioned looking for ways to take disparate comments from students on their web presences and aggregating them in a more unified manner for easier consumption by the teacher and other students (as opposed to subscribing to each and every student’s RSS feed, a task which can be onerous in classrooms larger than 20 people). To me this sounded like the general concept of a planet, and there are a few simple ways of accomplishing this already, specifically using RSS.
I was also thrilled to hear the ideas of POSSE and PESOS mentioned in such a setting!
An Invitation to Attendees
I’d invite those in attendance at the Domains 17 conference to visit not only the Indieweb wiki, but to feel free to actively participate in the on-going daily discussions (via IRC/Slack/Matrix/Web). I suspect that if there’s enough need/desire that the community would create a dedicated #education channel to help spur the effort to continue to push the idea of owning one’s own domain and using it for educational purposes out into the mainstream. The wiki pages and the always-on chat could be useful tools to help keep many of the educators and technologists who attended Domains17 not only connected after the event, but allow them to continue to push the envelope and document their progress for the benefit of others.
I’ll admit that one of my favorite parts of the Indieweb wiki is that it aggregates the work of hundreds of others in an intuitive way so that if I’m interested in a particular subject I can usually see the attempts that others have made (or at least links to them), determine what worked and didn’t for them, and potentially find the best solution for my particular use case. (I suspect that this is some of what’s missing in the “Domains” community at the moment based on several conversations I heard over the past several days.)
Tuesday on the NewsHour, a federal appeals court takes up President Trump's controversial immigration order. Also: Fact-checking the claim that the press underreports terror attacks, shocking details of a Syrian prison, how Betsy DeVos could reshape education policy, unique challenges for black children with autism and a new take on Timothy McVeigh's motivation for the Oklahoma City bombing.
The segment on autism in combination with the episode of Invisibilia on mental health I heard last night make me think we should drastically change how we treat and deal with mental health in our society.
The worst shame in the segment on autism was that the family felt shame for taking their son out into public.
Nice to see some of our favorite folks from NPR Radio making the rounds on television.
An exclusive look at data from the controversial web site Sci-Hub reveals that the whole world, both poor and rich, is reading pirated research papers.
Sci Hub has been in the news quite a bit over the past half a year and the bookmarked article here gives some interesting statistics. I’ll preface some of the following editorial critique with the fact that I love John Bohannon’s work; I’m glad he’s spent the time to do the research he has. Most of the rest of the critique is aimed at the publishing industry itself.
From a journalistic standpoint, I find it disingenuous that the article didn’t actually hyperlink to Sci Hub. Neither did it link out (or provide a full quote) to Alicia Wise’s Twitter post(s) nor link to her rebuttal list of 20 ways to access their content freely or inexpensively. Of course both of these are editorial related, and perhaps the rebuttal was so flimsy as to be unworthy of a link from such an esteemed publication anyway.
Sadly, Elsevier’s list of 20 ways of free/inexpensive access doesn’t really provide any simple coverage for graduate students or researchers in poorer countries which are the likeliest group of people using Sci Hub, unless they’re going to fraudulently claim they’re part of a class which they’re not, and is this morally any better than the original theft method? It’s almost assuredly never used by patients, which seem to be covered under one of the options, as the option to do so is painfully undiscoverable past their typical $30/paper firewalls. Their patchwork hodgepodge of free access is so difficult to not only discern, but one must keep in mind that this is just one of dozens of publishers a researcher must navigate to find the one thing they’re looking for right now (not to mention the thousands of times they need to do this throughout a year, much less a career).
Consider this experiment, which could be a good follow up to the article: is it easier to find and download a paper by title/author/DOI via Sci Hub (a minute) versus through any of the other publishers’ platforms with a university subscription (several minutes) or without a subscription (an hour or more to days)? Just consider the time it would take to dig up every one of 30 references in an average journal article: maybe just a half an hour via Sci Hub versus the days and/or weeks it would take to jump through the multiple hoops to first discover, read about, and then gain access and then download them from the over 14 providers (and this presumes the others provide some type of “access” like Elsevier).
Those who lived through the Napster revolution in music will realize that the dead simplicity of their system is primarily what helped kill the music business compared to the ecosystem that exists now with easy access through the multiple streaming sites (Spotify, Pandora, etc.) or inexpensive paid options like (iTunes). If the publishing business doesn’t want to get completely killed, they’re going to need to create the iTunes of academia. I suspect they’ll have internal bean-counters watching the percentage of the total (now apparently 5%) and will probably only do something before it passes a much larger threshold, though I imagine that they’re really hoping that the number stays stable which signals that they’re not really concerned. They’re far more likely to continue to maintain their status quo practices.
Some of this ease-of-access argument is truly borne out by the statistics of open access papers which are downloaded by Sci Hub–it’s simply easier to both find and download them that way compared to traditional methods; there’s one simple pathway for both discovery and download. Surely the publishers, without colluding, could come up with a standardized method or protocol for finding and accessing their material cheaply and easily?
“Hart-Davidson obtained more than 100 years of biology papers the hard way—legally with the help of the publishers. ‘It took an entire year just to get permission,’ says Thomas Padilla, the MSU librarian who did the negotiating.” John Bohannon in Who’s downloading pirated papers? Everyone
Personally, I use use relatively advanced tools like LibX, which happens to be offered by my institution and which I feel isn’t very well known, and it still takes me longer to find and download a paper than it would via Sci Hub. God forbid if some enterprising hacker were to create a LibX community version for Sci Hub. Come to think of it, why haven’t any of the dozens of publishers built and supported simple tools like LibX which make their content easy to access? If we consider the analogy of academic papers to the introduction of machine guns in World War I, why should modern researchers still be using single-load rifles against an enemy that has access to nuclear weaponry?
My last thought here comes on the heels of the two tweets from Alicia Wise mentioned, but not shown in the article:
She mentions that the New York Times charges more than Elsevier does for a full subscription. This is tremendously disingenuous as Elsevier is but one of dozens of publishers for which one would have to subscribe to have access to the full panoply of material researchers are typically looking for. Further, Elsevier nor their competitors are making their material as easy to find and access as the New York Times does. Neither do they discount access to the point that they attempt to find the subscription point that their users find financially acceptable. Case in point: while I often read the New York Times, I rarely go over their monthly limit of articles to need any type of paid subscription. Solely because they made me an interesting offer to subscribe for 8 weeks for 99 cents, I took them up on it and renewed that deal for another subsequent 8 weeks. Not finding it worth the full $35/month price point I attempted to cancel. I had to cancel the subscription via phone, but why? The NYT customer rep made me no less than 5 different offers at ever decreasing price points–including the 99 cents for 8 weeks which I had been getting!!–to try to keep my subscription. Elsevier, nor any of their competitors has ever tried (much less so hard) to earn my business. (I’ll further posit that it’s because it’s easier to fleece at the institutional level with bulk negotiation, a model not too dissimilar to the textbook business pressuring professors on textbook adoption rather than trying to sell directly the end consumer–the student, which I’ve written about before.)
(Trigger alert: Apophasis to come) And none of this is to mention the quality control that is (or isn’t) put into the journals or papers themselves. Fortunately one need’t even go further than Bohannon’s other writings like Who’s Afraid of Peer Review? Then there are the hordes of articles on poor research design and misuse of statistical analysis and inability to repeat experiments. Not to give them any ideas, but lately it seems like Elsevier buying the Enquirer and charging $30 per article might not be a bad business decision. Maybe they just don’t want to play second-banana to TMZ?
Interestingly there’s a survey at the end of the article which indicates some additional sources of academic copyright infringement. I do have to wonder how the data for the survey will be used? There’s always the possibility that logged in users will be indicating they’re circumventing copyright and opening themselves up to litigation.
I also found the concept of using the massive data store as a means of applied corpus linguistics for science an entertaining proposition. This type of research could mean great things for science communication in general. I have heard of people attempting to do such meta-analysis to guide the purchase of potential intellectual property for patent trolling as well.
Finally, for those who haven’t done it (ever or recently), I’ll recommend that it’s certainly well worth their time and energy to attend one or more of the many 30-60 minute sessions most academic libraries offer at the beginning of their academic terms to train library users on research tools and methods. You’ll save yourself a huge amount of time.
Math textbooks often seem difficult, obtuse, and often incomprehensible. Here are some hints and tips for making the situation better for all students.
Some General Advice for Math Students of All Ages
I recently saw the question “Why aren’t math textbooks more straightforward?” on Quora.
In fact, I would argue that most math textbooks are very straightforward!
The real issue most students are experiencing is one of relativity and experience. Mathematics is an increasingly sophisticated, cumulative, and more complicated topic the longer you study it. Fortunately, over time, it also becomes easier, more interesting, and intriguingly more beautiful.
As an example of what we’re looking at and what most students are up against, let’s take the topic of algebra. Typically in the United States one might take introductory algebra in eighth grade before taking algebra II in ninth or tenth grade. (For our immediate purposes, here I’m discounting the potential existence of a common pre-algebra course that some middle schools, high schools, and even colleges offer.) Later on in college, one will exercise one’s algebra muscles in calculus and may eventually get to a course called abstract algebra as an upper-level undergraduate (in their junior or senior years). Most standard undergraduate abstract algebra textbooks will cover ALL of the material that was in your basic algebra I and algebra II texts in about four pages and simply assume you just know the rest! Naturally, if you started out with the abstract algebra textbook in eighth grade, you’d very likely be COMPLETELY lost. This is because the abstract algebra textbook is assuming that you’ve got some significant prior background in mathematics (what is often referred to in the introduction to far more than one mathematics textbook as “mathematical sophistication”, though this phrase also implicitly assumes knowledge of what a proof is, what it entails, how it works, and how to actually write one).
Following the undergraduate abstract algebra textbook there’s even an additional graduate level course (or four) on abstract algebra (or advanced subtopics like group theory, ring theory, field theory, and Galois theory) that goes into even more depth and subtlety than the undergraduate course; the book for this presumes you’ve mastered the undergraduate text and goes on faster and further.
A Weightlifting Analogy
To analogize things to something more common, suppose you wanted to become an Olympic level weightlifter. You’re not going to go into the gym on day one and snatch and then clean & jerk 473kg! You’re going to start out with a tiny fraction of that weight and practice repeatedly for years slowly building up your ability to lift bigger and bigger weights. More likely than not, you’ll also very likely do some cross-training by running, swimming, and even lifting other weights to strengthen your legs, shoulder, stomach, and back. All of this work may eventually lead you to to win the gold medal in the Olympics, but sooner or later someone will come along and break your world record.
Mathematics is certainly no different: one starts out small and with lots of work and practice over time, one slowly but surely ascends the rigors of problems put before them to become better mathematicians. Often one takes other courses like physics, biology, and even engineering courses that provide “cross-training.” Usually when one is having issues in a math class it’s because they’re either somehow missing something that should have come before or because they didn’t practice enough in their prior classwork to really understand all the concepts and their subtleties. As an example, the new material in common calculus textbooks is actually very minimal – the first step in most problems is the only actual calculus and the following 10 steps are just practicing one’s algebra skills. It’s usually in carrying out the algebra that one makes more mistakes than in the actual calculus.
Often at the lower levels of grade-school mathematics, some students can manage to just read a few examples and just seem to “get” the answers without really doing a real “work out.” Eventually they’ll come to a point at which they hit a wall or begin having trouble, and usually it comes as the result of not actually practicing their craft. One couldn’t become an Olympic weightlifter by reading books about weightlifting, they need to actually get in the gym and workout/practice. (Of course, one of the places this analogy breaks down is that weightlifting training is very linear and doesn’t allow one to “skip around” the way one could potentially in a mathematics curriculum.)
I’m reminded of a quote by mathematician Pierre Anton Grillet: “…algebra is like French pastry: wonderful, but cannot be learned without putting one’s hands to the dough.” It is one of the most beautiful expressions of the recurring sentiment written by almost every author into the preface of nearly every mathematics text at or above the level of calculus. They all exhort their students to actually put pencil to paper and work through the logic of their arguments and the exercises to learn the material and gain some valuable experience. I’m sure that most mathematics professors will assure you that in the end, only a tiny fraction of their students actually do so. Some of the issue is that these exhortations only come in textbooks traditionally read at the advanced undergraduate level, when they should begin in the second grade.
“It’s Easy to See”
A common phrase in almost every advanced math textbook on the planet is the justification, “It’s easy to see.” The phrase, and those like it, should be a watchword for students to immediately be on their guard! The phrase is commonly used in proofs, discussions, conversations, and lectures in which an author or teacher may skip one or more steps which she feels should be obvious to her audience, but which, in fact, are far more commonly not obvious.
It’s become so cliche that some authors actually mention specifically in their prefaces that they vow not to use the phrase, but if they do so, they usually let slip some other euphemism that is its equivalent.
The problem with the phrase is that everyone, by force of their own circumstances and history, will view it completely differently. A step that is easy for someone with a Ph.D. who specialized in field theory to “see” may be totally incomprehensible for a beginning student of algebra I in the same way that steps that were easy for Girgory Perelman to see in his proof of the Poincaré conjecture were likewise completely incomprehensible for teams of multiple tenured research professors of mathematics to see. (cross reference: The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O’Shea (Walker & Co., 2007))
How to Actively Read a Math Text
So how are students to proceed? It will certainly help to see a broader road map of what lies ahead and what the expected changes in terrain will look like. It will also help greatly if students have a better idea how to approach mathematics for themselves and even by themselves in many cases.
In my opinion, the most common disconnect occurs somewhere between high school mathematics and early college mathematics (usually a calculus sequence, linear algebra) and then again between linear algebra/differential equations (areas which usually have discussion followed by examples and then crank-out problems) and higher abstract mathematical areas like analysis, abstract algebra, topology (areas in which the definition-theorem-proof cycle of writing is more common and seemingly more incomprehensible to many).
The first big issue in early college mathematics is the increased speed at which college courses move. Students used to a slower high school pace where the teachers are usually teaching to the middle or lower end of the class get caught unaware as their college professors teach to the higher ability students and aren’t as afraid to leave the lower end of the spectrum behind. Just like high school athletes are expected to step up their game when they make the transition to college and similarly college athletes who go pro, mathematics students should realize they’re expected to step up their game at the appropriate times.
Often math students (and really any student of any subject) relies on the teacher assigning readings or problems from their book rather than excersizing their curiosity to more avidly and broadly explore the material on their own. If they can take the guidance of their teacher as well as that of the individual authors of books, they may make it much further on their own. High school teachers often skip sections of textbooks for time, but students should realize that there is profitable and interesting material that they’re skipping. Why not go and read it on their own?
Earlier I mentioned that an average undergraduate abstract algebra textbook might cover the totality of a high school algebra textbook in about three pages. What does this mean for upper level mathematics students? It almost always means that the density of material in these books is far greater than that of their earlier textbooks. How is this density arrived at? Authors of advanced textbooks leave out far more than they’re able to put in, otherwise their 300 page textbooks, if written at the same basic level as those that came before would be much more ponderous 1000+ page textbooks. What are they leaving out? Often they’re leaving out lots of what might be useful discussion, but more often, they’re leaving out lots of worked out examples. For example, a high school text will present a definition or concept and then give three or more illustrations or examples of problems relating to the concept. The exercises will then give dozens of additional drill problems to beat the concept to death. This type of presentation usually continues up to the level of calculus where one often sees massive tomes in the 800+ page length. Math texts after this point generally don’t go much over 300 pages as a rule, and it’s primarily because they’re leaving the examples out of the proverbial equation.
How does one combat this issue? Students need to more actively think back to the math they’ve taken previously and come up with their own simple examples of problems, and work though them on their own. Just because the book doesn’t give lots of examples doesn’t mean that they don’t exist.
In fact, many textbooks are actually presenting examples, they’re just hiding them with very subtle textual hints. Often in the presentation of a concept, the author will leave out one or more steps in a proof or example and hint to the student that they should work through the steps themselves. (Phrases like: “we leave it to the reader to verify” or “see example 2.”) Sometimes this hint comes in the form of that dreaded phrase, “It’s easy to see.” When presented with these hints, it is incumbent (or some students may prefer the word encumbering) on the student to think through the missing steps or provide the missing material themselves.
While reading mathematics, students should not only be reading the words and following the steps, but they should actively be working their way through all of the steps (missing or not) in each of the examples or proofs provided. They must read their math books with pencil and paper in hand instead of the usual format of reading their math book and then picking up paper and pencil to work out problems afterwards. Most advanced math texts suggest half a dozen or more problems to work out within the text itself before presenting a dozen or more additional problems usually in a formal section entitled “Exercises”. Students have to train themselves to be thinking about and working out the “hidden” problems within the actual textual discussion sections.
Additionally, students need to consider themselves “researchers” or think of their work as discovery or play. Can they come up with their own questions or exercises that relate the concepts they’ve read about to things they’ve done in the past? Often asking the open ended question, “What happens if I…” can be very useful. One has to imagine that this is the type of “play” that early mathematicians like Euclid, Gauss, and Euler did, and I have to say, this is also the reason that they discovered so many interesting properties within mathematics. (I always like to think that they were the beneficiaries of “picking the lowest hanging fruit” within mathematics – though certainly they discovered some things that took some time to puzzle out; we take some of our knowledge for granted as sitting on the shoulders of giants does allow us to see much further than we could before.)
As a result of this newly discovered rule, students will readily find that while they could read a dozen pages of their high school textbooks in just a few minutes, it may take them between a half an hour to two hours to properly read even a single page of an advanced math text. Without putting in this extra time and effort they’re going to quickly find themselves within the tall grass (or, more appropriately weeds).
Another trick of advanced textbooks is that, because they don’t have enough time or space within the primary text itself, authors often “hide” important concepts, definitions, and theorems within the “exercises” sections of their books. Just because a concept doesn’t appear in the primary text doesn’t mean it isn’t generally important. As a result, students should always go out of their way to at least read through all of the exercises in the text even if they don’t spend the time to work through them all.
One of the difficult things about advanced abstract mathematics is that it is most often very cumulative and even intertwined, so when one doesn’t understand the initial or early portions of a textbook, it doesn’t bode well for the later sections which require one to have mastered the previous work. This is even worse when some courses build upon the work of earlier courses, so for example, doing well in calculus III requires that one completely mastered calculus I. At some of the highest levels like courses in Lie groups and Lie algebras requires that one mastered the material in multiple other prior courses like analysis, linear algebra, topology, and abstract algebra. Authors of textbooks like these will often state at the outset what material they expect students to have mastered to do well, and even then, they’ll often spend some time giving overviews of relevant material and even notation of these areas in appendices of their books.
As a result of this, we can take it as a general rule: “Don’t ever skip anything in a math textbook that you don’t understand.” Keep working on the concepts and examples until they become second nature to you.
Finally, more students should think of mathematics as a new language. I’ve referenced the following Galileo quote before, but it bears repeating (emphasis is mine):
Though mathematical notation has changed drastically (for the better, in my opinion) since Galileo’s time, it certainly has its own jargon, definitions, and special notations. Students should be sure to spend some time familiarizing themselves with current modern notation, and especially the notation in the book that they choose. Often math textbooks will have a list of symbols and their meanings somewhere in the end-papers or the appendices. Authors usually go out of their way to introduce notation somewhere in either the introduction, preface, appendices, or often even in an introductory review chapter in which they assume most of their students are very familiar with, but they write it anyway to acclimate students to the particular notation they use in their text. This notation can often seem excessive or even obtuse, but generally it’s very consistent across disciplines within mathematics, but it’s incredibly useful and necessary in making often complex concepts simple to think about and communicate to others. For those who are lost, or who want help delving into areas of math seemingly above their heads, I highly recommend the text Mathematical Notation: A Guide for Engineers and Scientists by my friend Edward R. Scheinerman as a useful guide.
A high school student may pick up a textbook on Lie Groups and be astounded at the incomprehensibility of the subject, but most of the disconnect is in knowing and understanding the actual language in which the text is written. A neophyte student of Latin would no sooner pick up a copy of Cicero and expect to be able to revel in the beauty and joy of the words or their meaning without first spending some time studying the vocabulary, grammar, and syntax of the language. Fortunately, like Latin, once one has learned a good bit of math, the notations and definitions are all very similar, so once you can read one text, you’ll be able to appreciate a broad variety of others.
Actively Reading a Mathematics Text Review:
Work through the steps of everything within the text
Come up with your own examples
Work through the exercises
Read through all the exercises, especially the ones that you don’t do
Don’t ever skip anything you don’t fully understand
Math is a language: spend some time learning (memorizing) notation
Naturally there are exceptions to the rule. Not all mathematics textbooks are great, good, or even passable. There is certainly a spectrum of textbooks out there, and there are even more options at the simpler (more elementary) end, in part because of there is more demand. For the most part, however, most textbooks are at least functional. Still one can occasionally come across a very bad apple of a textbook.
Because of the economics of textbook publishing, it is often very difficult for a textbook to even get published if it doesn’t at least meet a minimum threshold of quality. The track record of a publisher can be a good indicator of reasonable texts. Authors of well-vetted texts will often thank professors who have taught their books at other universities or even provide a list of universities and colleges that have adopted their texts. Naturally, just because 50 colleges have adopted a particular text doesn’t necessarily mean that that it is necessarily of high quality.
One of the major issues to watch out for is using the textbook written by one’s own professor. While this may not be an issue if your professor is someone like Serge Lang, Gilbert Strang, James Munkres, Michael Spivak, or the late Walter Rudin, if your particular professor isn’t supremely well known in his or her field, is an adjunct or associate faculty member, or is a professor at a community college, then: caveat emptor.
Since mathematics is a subject about clear thinking, analysis, and application of knowledge, I recommend that students who feel they’re being sold a bill of goods in their required/recommended textbook(s), take the time to look at alternate textbooks and choose one that is right for themselves. For those interested in more on this particular sub-topic I’ve written about it before: On Choosing Your own Textbooks.
Often, even with the best intentions, some authors can get ahead of themselves or the area at hand is so advanced that it is difficult to find a way into it. As an example, we might consider Lie groups and algebras, which is a fascinating area to delve into. Unfortunately it can take several years of advanced work to get to a sufficient level to even make a small dent into any of the textbooks in the area, though some research will uncover a handful of four textbooks that will get one quite a way into the subject with a reasonable background in just analysis and linear algebra.
When one feels like they’ve hit a wall, but still want to struggle to succeed, I’m reminded of the advice of revered mathematical communicator Paul Halmos, whose book Measure Theory needed so much additional background material, that instead of beginning with the traditional Chapter 1, he felt it necessary to include a Chapter 0 (he actually called his chapters “sections” in the book) and even then it had enough issueshewas cornered into writing the statement:
This is essentially the mathematician’s equivalent of the colloquialism “Fake it ’til you make it.”
When all else fails, use this adage, and don’t become discouraged. You’ll get there eventually!
The Postdoctoral Experience Revisited builds on the 2000 report Enhancing the Postdoctoral Experience for Scientists and Engineers. That ground-breaking report assessed the postdoctoral experience and provided principles, action points, and recommendations to enhance that experience. Since the publication of the 2000 report, the postdoctoral landscape has changed considerably. The percentage of PhDs who pursue postdoctoral training is growing steadily and spreading from the biomedical and physical sciences to engineering and the social sciences. The average length of time spent in postdoctoral positions seems to be increasing. The Postdoctoral Experience Revisited reexamines postdoctoral programs in the United States, focusing on how postdocs are being guided and managed, how institutional practices have changed, and what happens to postdocs after they complete their programs. This book explores important changes that have occurred in postdoctoral practices and the research ecosystem and assesses how well current practices meet the needs of these fledgling scientists and engineers and of the research enterprise.
The Postdoctoral Experience Revisited takes a fresh look at current postdoctoral fellows - how many there are, where they are working, in what fields, and for how many years. This book makes recommendations to improve aspects of programs - postdoctoral period of service, title and role, career development, compensation and benefits, and mentoring. Current data on demographics, career aspirations, and career outcomes for postdocs are limited. This report makes the case for better data collection by research institution and data sharing.
A larger goal of this study is not only to propose ways to make the postdoctoral system better for the postdoctoral researchers themselves but also to better understand the role that postdoctoral training plays in the research enterprise. It is also to ask whether there are alternative ways to satisfy some of the research and career development needs of postdoctoral researchers that are now being met with several years of advanced training. Postdoctoral researchers are the future of the research enterprise. The discussion and recommendations of The Postdoctoral Experience Revisited will stimulate action toward clarifying the role of postdoctoral researchers and improving their status and experience.
The National Academy of Sciences has published a (free) book: The Postdoctoral Experience (Revisited) discussing where we’re at and some ideas for a way forward.
Most might agree that our educational system is far less than ideal, but few pay attention to significant problems at the highest levels of academia which are holding back a great deal of our national “innovation machinery”. The National Academy of Sciences has published a (free) book: The Postdoctoral Experience (Revisited) discussing where we’re at and some ideas for a way forward. There are some interesting ideas here, but we’ve still got a long way to go.
I recently ran across this TED talk and felt compelled to share it. It really highlights some of my own personal thoughts on how science should be taught and done in the modern world. It also overlaps much of the reading I’ve been doing lately on innovation and creativity. If these don’t get you to watch, then perhaps mentioning that Alon manages to apply comedy and improvisation techniques to science will.
Uri Alon was already one of my scientific heroes, but this adds a lovely garnish.
I’ve been a proponent and user of a variety of mnemonic systems since I was about eleven years old. The two biggest and most useful in my mind are commonly known as the “method of loci” and the “major system.” The major system is also variously known as the phonetic number system, the phonetic mnemonic system, or Hergione’s mnemonic system after French mathematician and astronomer Pierre Hérigone (1580-1643) who is thought to have originated its use.
The major system generally works by converting numbers into consonant sounds and then from there into words by adding vowels under the overarching principle that images (of the words) can be remembered more easily than the numbers themselves. For instance, one could memorize one’s grocery list of a hundred items by associating each shopping item on a numbered list with the word associated with the individual number in the list. As an example, if item 22 on the list is lemons, one could translate the number 22 as “nun” within the major system and then associate or picture a nun with lemons – perhaps a nun in full habit taking a bath in lemons to make the image stick in one’s memory better. Then at the grocery store, when going down one’s list, when arriving at number 22 on the list, one automatically translates the number 22 to “nun” which will almost immediately conjure the image of a nun taking a bath in lemons which gives one the item on the list that needed to be remembered. This comes in handy particularly when one needs to be able to remember large lists of items in and out of order.
The following generalized chart, which can be found in a hoard of books and websites on the topic, is fairly canonical for the overall system:
Mnemonic for remembering the numeral and consonant relationship
s, z, soft c
“z” is the first letter of zero; the other letters have a similar sound
t & d have one downstroke and sound similar (some variant systems include “th”)
n has two downstrokes
m has three downstrokes; m looks like a “3” on its side
last letter of four; 4 and R are almost mirror images of each other
L is the Roman Numeral for 50
/ʃ/ /ʒ/ /tʃ/ /dʒ/
j, sh, soft g, soft “ch”
a script j has a lower loop; g is almost a 6 rotated
k, hard c, hard g, hard “ch”, q, qu
capital K “contains” two sevens (some variant systems include “ng”)
script f resembles a figure-8; v sounds similar (v is a voiced f)
p is a mirror-image 9; b sounds similar and resembles a 9 rolled around
Vowel sounds, w,h,y
w and h are considered half-vowels; these can be used anywhere without changing a word’s number value
There are a variety of ways to use the major system as a code in addition to its uses in mnemonic settings. When I was a youth, I used it to write coded messages and to encrypt a variety of things for personal use. After I had originally read Dr. Bruno Furst’s series of booklets entitled You Can Remember: A Home Study Course in Memory and Concentration  , I had always wanted to spend some time creating an alternate method of writing using the method. Sadly I never made the time to do the project, but yesterday I made a very interesting discovery that, to my knowledge, doesn’t seem to have been previously noticed!
My discovery began last week when I read an article in The Atlantic by journalist Dennis Hollier entitled How to Write 225 Words Per Minute with a Pen: A Lesson in the Lost Technology of Shorthand.  In the article, which starts off with a mention of the Livescribe pen – one of my favorite tools, Mr. Hollier outlines the use of the Gregg System of Shorthand which was invented by John Robert Gregg in 1888. The description of the method was intriguing enough to me that I read a dozen additional general articles on shorthand on the internet and purchased a copy of Louis A. Leslie’s two volume text Gregg Shorthand: Functional Method. 
I was shocked, on page x of the front matter, just before the first page of the text, to find the following “Alphabet of Gregg Shorthand”:
Gregg Shorthand is using EXACTLY the same consonant-type breakdown of the alphabet as the major system!
Apparently I wasn’t the first to have the idea to turn the major system into a system of writing. The fact that the consonant breakdowns for the major system coincide almost directly to those for the shorthand method used by Gregg cannot be a coincidence!
The Gregg system works incredibly well precisely because the major system works so well. The biggest difference between the two systems is that Gregg utilizes a series of strokes (circles and semicircles) to indicate particular vowel sounds which allows for better differentiation of words which the major system doesn’t generally take into consideration. From an information theoretic standpoint, this is almost required to make the coding from one alphabet to the other possible, but much like ancient Hebrew, leaving out the vowels doesn’t remove that much information. Gregg, also like Hebrew, also uses dots and dashes above or below certain letters to indicate the precise sound of many of its vowels.
The upside of all of this is that the major system is incredibly easy to learn and use, and from here, learning Gregg shorthand is just a hop, skip , and a jump – heck, it’s really only just a hop because the underlying structure is so similar. Naturally as with the major system, one must commit some time to practicing it to improve on speed and accuracy, but the general learning of the system is incredibly straightforward.
Because the associations between the two systems are so similar, I wasn’t too surprised to find that some of the descriptions of why certain strokes were used for certain letters were very similar to the mnemonics for why certain letters were used for certain numbers in the major system.
One thing I have noticed in my studies on these topics is the occasional references to the letter combinations “NG” and “NK”. I’m curious why these are singled out in some of these systems? I have a strong suspicion that their inclusion/exclusion in various incarnations of their respective systems may be helpful in dating the evolution of these systems over time.
I’m aware that various versions of shorthand have appeared over the centuries with the first recorded having been the “Tironian Notes” of Marcus Tullius Tiro (103-4 BCE) who apparently used his system to write down the speeches of his master Cicero. I’m now much more curious at what point the concepts for shorthand and the major system crossed paths or converged? My assumption would be that it happened in the late Renaissance, but it would be nice to have the underlying references and support for such a timeline. Perhaps it was with Timothy Bright’s publication of Characterie; An Arte of Shorte, Swifte and Secrete Writing by Character (1588)  , John Willis’s Art of Stenography (1602)  , Edmond Willis’s An abbreviation of writing by character (1618)  , or Thomas Shelton’s Short Writing (1626)  ? Shelton’s system was certainly very popular and well know because it was used by both Samuel Pepys and Sir Isaac Newton.
Certainly some in-depth research will tell, though if anyone has ideas, please don’t hesitate to indicate your ideas in the comments.
UPDATE on 7/6/14:
I’m adding a new chart making the correspondence between the major system and Gregg Shorthand more explicit.
B. Furst, You Can Remember: A Home Study Course in Memory and Concentration. Markus-Campbell Co., 1965.