The main purpose of this blog is to share updates about the open-access, open-source textbook Understanding Linear Algebra. Though work is continuing on this project, the HTML version of the text is now freely available, the forthcoming PDF version will also be free, and low-cost print options will be provided. The PreTeXt source code will be posted on GitHub as well.
My awesome colleague @davidaustinm is unveiling his new, open-source linear algebra text at the JMM, but you can access it NOW at his (new!) blog, the aptly named "More Linear Algebra": https://t.co/AAreqGk8DW
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?
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."
Michelle Pacansky-Brock says digital learning is reshaping the higher ed landscape, and suggests five things instructors need to succeed.
As online and blended learning reshape the landscape of teaching and learning in higher education, the need increases to encourage and support faculty in moving from delivering passive, teacher-centered experiences to designing active, student-centered learning.
Our new social era is rich with simple, free to low-cost emerging technologies that are increasing experimentation and discovery in the scholarship of teaching and learning. While the literature about Web 2.0 tools impacting teaching and learning is increasing, there is a lack of knowledge about how the adoption of these technologies is impacting the support needs of higher education faculty. This knowledge is essential to develop new, sustainable faculty support solutions.
This might make an outline for a nice book, but as an article it’s a bit wonkish and doesn’t get into the meat of much.
Feel free to either subscribe to the list (useful when adding streams to things like Tweetdeck), or for quickly scanning down the list and following people on a particular topic en-masse. Hopefully it will help people to remain connected following the conference. I’ve written about some other ideas about staying in touch here.
If you or someone you know is conspicuously missing, please let me know and I’m happy to add them. Hopefully this list will free others from spending the inordinate amount of time to create similar bulk lists from the week.
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.)
Seymour Papert’s Mindstorms was published by Basic Books in 1980, and outlines his vision of children using computers as instruments for learning. A second edition, with new Forewords by John Sculley and Carol Sperry, was published in 1993. The book remains as relevant now as when first published almost forty years ago.
The Media Lab is grateful to Seymour Papert’s family for allowing us to post the text here. We invite you to add your comments and reflections.
If you are interested in purchasing the print edition of Mindstorms, please visit Basic Books.
Great to see this interview with my friend and mathematician Richard Brown from Johns Hopkins Unviersity. Psst: He’s got an interesting little blog, or you can follow some of his work on Facebook and Twitter.
The beginning of a four part series in which I provide a gradation of books and texts that lie in the intersection of the application of information theory, physics, and engineering practice to the area of biology.
Previously, I had made a large and somewhat random list of books which lie in the intersection of the application of information theory, physics, and engineering practice to the area of biology. Below I’ll begin to do a somewhat better job of providing a finer gradation of technical level for both the hobbyist or the aspiring student who wishes to bring themselves to a higher level of understanding of these areas. In future posts, I’ll try to begin classifying other texts into graduated strata as well. The final list will be maintained here: Books at the Intersection of Information Theory and Biology.
Introductory / General Readership / Popular Science Books
These books are written on a generally non-technical level and give a broad overview of their topics with occasional forays into interesting or intriguing subtopics. They include little, if any, mathematical equations or conceptualization. Typically, any high school student should be able to read, follow, and understand the broad concepts behind these books. Though often non-technical, these texts can give some useful insight into the topics at hand, even for the most advanced researchers.
One of the best books on the concept of entropy out there. It can be read even by middle school students with no exposure to algebra and does a fantastic job of laying out the conceptualization of how entropy underlies large areas of the broader subject. Even those with Ph.D.’s in statistical thermodynamics can gain something useful from this lovely volume.
A relatively recent popular science volume covering various conceptualizations of what information is and how it’s been dealt with in science and engineering. Though it has its flaws, its certainly a good introduction to the beginner, particularly with regard to history.
The four books above have a significant amount of overlap. Though one could read all of them, I recommend that those pressed for time choose Ben-Naim first. As I write this I’ll note that Ben-Naim’s book is scheduled for release on May 30, 2015, but he’s been kind enough to allow me to read an advance copy while it was in process; it gets my highest recommendation in its class. Loewenstein covers a bit more than Avery who also has a more basic presentation. Most who continue with the subject will later come across Yockey’s Information Theory and Molecular Biology which is similar to his text here but written at a slightly higher level of sophistication. Those who finish at this level of sophistication might want to try Yockey third instead.
In the coming weeks/months, I’ll try to continue putting recommended books on the remainder of the rest of the spectrum, the balance of which follows in outline form below. As always, I welcome suggestions and recommendations based on others’ experiences as well. If you’d like to suggest additional resources in any of the sections below, please do so via our suggestion box. For those interested in additional resources, please take a look at the ITBio Resources page which includes information about related research groups; references and journal articles; academic, research institutes, societies, groups, and organizations; and conferences, workshops, and symposia.
Lower Level Undergraduate
These books are written at a level that can be grasped and understood by most with a freshmen or sophomore university level. Coursework in math, science, and engineering will usually presume knowledge of calculus, basic probability theory, introductory physics, chemistry, and basic biology.
Upper Level Undergraduate
These books are written at a level that can be grasped and understood by those at a junior or senor university level. Coursework in math, science, and engineering may presume knowledge of probability theory, differential equations, linear algebra, complex analysis, abstract algebra, signal processing, organic chemistry, molecular biology, evolutionary theory, thermodynamics, advanced physics, and basic information theory.
These books are written at a level that can be grasped and understood by most working at the level of a master’s level at most universities. Coursework presumes all the previously mentioned classes, though may require a higher level of sub-specialization in one or more areas of mathematics, physics, biology, or engineering practice. Because of the depth and breadth of disciplines covered here, many may feel the need to delve into areas outside of their particular specialization.
Imagine you had to take an art class in which you were taught how to paint a fence or a wall, but you were never shown the paintings of the great masters, and you weren't even told that such paintings existed. Pretty soon you'd be asking, why study art?
That's absurd, of course, but it's surprisingly close to the way we teach children mathematics. In elementary and middle school and even into high school, we hide math's great masterpieces from students' view. The arithmetic, algebraic equations and geometric proofs we do teach are important, but they are to mathematics what whitewashing a fence is to Picasso — so reductive it's almost a lie.
Most of us never get to see the real mathematics because our current math curriculum is more than 1,000 years old. For example, the formula for solutions of quadratic equations was in al-Khwarizmi's book published in 830, and Euclid laid the foundations of Euclidean geometry around 300 BC. If the same time warp were true in physics or biology, we wouldn't know about the solar system, the atom and DNA. This creates an extraordinary educational gap for our kids, schools and society.
An interesting train of thought to be sure. I should post in response to this, or at least think about how it could be structured. I definitely want to come back to write more about this topic.
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!