One of my students asked me about the origin of this symbol, and I had to confess I didn’t know. I always figured it was somehow related to the much more common capital Greek delta, \( \Delta \), but the real story is a lot more colorful than that.

The nabla is so-called because it looks like a harp; the Greek word for the Hebrew or Egyptian form of a harp is “nabla” . What does a harp have to do with mathematics? The image came up in relation to mathematics through part of a conversation between Peter Guthrie Tait and James Clerk Maxwell:

The name was used in playful intercourse between Tait and Clerk Maxwell, who in a letter of uncertain date finished a brief sketch of a particular problem in orthogonal surfaces by the remark “It is neater and perhaps wiser to compose a nablody on this theme which is well suited for this species of composition.” [...]

It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing the word earlier than he did. The one published use of the word by Maxwell is in the title to his homorous Tyndallic Ode, which is dedicated to the “Chief Musician upon Nabla,” that is, Tait.

In a letter from Maxwell to Tait on Nov. 7, 1870, Maxwell wrote, ”What do you call this? Atled?”

In a letter from Maxwell to Tait on Jan. 23, 1871, Maxwell began with, ”Still harping on that Nabla?”

(The above quote is from this text file that contains a lot more on this subject.)

I like that term “atled” — that’s the backwards spelling of “delta” of course. I keep thinking I’ve seen “atled” as an alternative \( \LaTeX \) command for nabla.

It’s also nice to see two giants in mathematics have a bit of fun together as they developed some of their mathematics. The linked article mentions that other mathematicians apparently more stuffy than Tait and Maxwell, were reluctant to adopt this symbol and its name because it didn’t seem serious enough. The joke’s on them!

Here’s an excerpt:

The Bible describes the early universe as “without form and void” (Genesis 1:2). This is certainly true, according to Marcus Chown, in terms of how much information was in the universe then. Expanding on the ideas of physicist Stephen Hsu, Chown argues that, just after the Big Bang, the universe contained no more than 200 bytes of information. By comparison, the first two sentences of this article contain 190 bytes of information. A universe with so little content would certainly seem “void.” But today, the amount of information just within arms’ reach—to say nothing of the rest of the universe—seems infinite. Where did all that information come from?

My piece goes on to discuss how random processes introduced information into the universe, the relationship between randomness and information content, and what the notion of ubiquitous randomness means when thinking about an ultimate purpose in the universe.

I’ve been a fan of Books & Culture for a long time, and it was a real thrill to be able to contribute something. I hope you find it interesting.

Certainly in the inverted classroom model and in many instances of peer instruction, it’s simply not the case that “there is no lecture”. There can be a lot of lecture in either of these models. It’s just that the lectures are not given in class. They are broken up into rewindable, pause-able, digestible chunks and posted online where people can view them on their own schedules and according to their own listening practices. For my inverted MATLAB class, I recorded 41 screencasts’ worth of lectures, amounting to about 332 minutes of lecture, which is not very far off from what you would get in class over the course of a semester in a one-credit course (which this was).

So if there is roughly the same amount of lecture in this kind of class as there is in a traditional classroom, and the lectures are over the same material and given in more or less the same way as they would be if they were given in class, why is there the sense that

• If the teacher is lecturing me during class, I am being taught, but
• If the lectures are outside of class, I am teaching myself?

To my way of thinking, it’s in the traditional classroom where students are teaching themselves, because the real learning comes from assimilation — not receiving information, but internalizing it through working on problems. And in the traditional classroom, I am given this work to do on my own. I may have the option to work in groups or come to office hours, but that’s not the default. It may not even be an option.

Where is the belief that “lectures outside of class = teaching myself” coming from? And is this an accurate way to sum up the experience that students have?

Image: http://www.flickr.com/photos/thowi/

• Modeling thought processes. The benefit of hearing an expert learner lecture on a subject is that, if the lecture is clearly given, the audience can gain some insights into what makes the expert an expert. A good lecture does more than convey facts or put problems on the board — it lays bare the cognitive processes that an expert uses to assimilate those facts or think his or her way through those problems.
• Sharing cognitive structures. Lectures provide the important opportunity for the lecturer to share the mental models and internal cognitive frameworks that worked for him/her when he/she was learning the content. For example, when I took Calculus as a high schooler, I learned the Quotient Rule using the little ditty “Ho D Hi minus Hi D Ho over Ho Ho” and I still cannot perform the Quotient Rule by hand without singing that to myself. I share that whenever I teach Calculus and it works with students — and it’s not something they would necessarily have come up with on their own.
• Giving context. Good lecturers know more than just their subject material. They know the context in which that content sits and how the material relates to other things — things that a novice learner might not think about, just because he or she is a novice. Lectures are good places to learn some things from people with a broader set of experiences than you have.
• Telling stories. Stories from popular history or culture or from the professor’s own life are a kind of cognitive structure that help students to relate to the course and see the course content in a different way. For example, students learning logic have trouble with the notion that a conditional (“if-then”) statement is actually considered true if the hypothesis (the “if” part) is false. So I tell a story about promising my kids ice cream if they finish their dinner. If they didn’t finish their dinner but I got them ice cream anyway, it doesn’t make me a liar — so my “conditional statement” was still true. This connects somehow where truth tables don’t.

Notice that what I don’t include in this list is the one thing that lectures seem most commonly used for: information transfer. In fact lectures, while effective at “covering material”, are terrible for information transfer from the student’s point of view. There are serious problems with retention and recall of information given in a lecture even if the lecture is rhetorically solid — and this is to say nothing about the disconnect between the length of the average lecture and the average human being’s attention span. Resorting to a lecture because I need to “cover material” is just an admission that I didn’t design my course well. If that’s all the lecture is for, put it online so students can at least pause and rewind.

Notice also that I do not count whether a lecture is inspiring or not. No doubt many lectures are inspiring, but being inspired and being taught are not the same thing, and just having one’s thoughts provoked doesn’t mean that one has interacted with the lecturer in any real way. I am inspired by many of the TED talks and sermons I hear, but it doesn’t mean I have learned anything. I had a pastor once who was unfailingly inspirational — and I couldn’t remember a single bit of what he preached on, nor could I give even a coherent outline of the sermon, within two hours of church being over.

But while running an entire class in the lecture format is probably not best for students, lectures do have their place, and when it makes sense to give one, we should do so with clarity, organization, and rhetorical skill. We ought to aim for balance, with lecture and active work combining together to produce as rich a  learning environment as possible.

Image: http://www.flickr.com/photos/kaptainkobold/

I’ve been an end-user of SoTL for a long time and have done a lot of you might call “scholarship” in SoTL — for example all the writing and speaking I’ve done about the inverted classroom and clickers — but I’ve not done what I consider actual research in SoTL. One of the reasons I came to GVSU was to have the time, space, and support to transition into doing research in SoTL, and this faculty learning community is the first big step in that direction.

So now my next step is to think about what kinds of questions I would like eventually to turn into research projects. I need some time to really process these, and especially to soak in the existing literature and see what other kinds of questions are out there, but modulo those concerns, here’s a short list of ideas:

• There’s not a lot of actual research on the inverted classroom out there, and that’s a real problem for inverted classroom advocates like me. Just about any claim that one can find about the benefits of this model that you can find on my blog are candidates for a research study. For example, does the inverted classroom really use study time more efficiently than a standard classroom? Does the inverted classroom change the way students view their capacity to learn things on their own?
• For that matter, let’s cut to the chase: Does the inverted classroom do a better job of engendering student learning than its traditional counterparts? That’s a loaded and ill-defined question — which is part of the problem.
• Also, what elements need to be in place for an inverted classroom to be effective? For example, what do the screencasts need to look like (in terms of length, style, etc.)? What are the best approaches for creating guided practice problems? What course assessments work best?
• Apart from the inverted classroom, there’s a lot of room to do research on peer instruction in mathematics. Basically take anything that Mazur studied and see if it can be replicated in a calculus (for instance) class. One of the big sticking points here right now is that there is no “Force Concept Inventory” [PDF] for calculus* — no instrument for measuring baseline skills and therefore normalized gains in learning arising from peer instruction.
• It turns out my colleague from the Computer Science department has been working with clickers and peer instruction for several years now. We’re already making plans to visit each others’ classes. This is one of the real strengths of faculty communities on a single campus — the guy down the hall who you’d otherwise probably never see turns out to be a potential collaborator.

This short, unrefined list probably doesn’t contain any well-defined research questions. Part of my problem is that I don’t yet really know what a well-defined research question in SoTL even looks like. But I figure, incompetence never stopped me before, so why start worrying about it now? I’m just excited to get moving in the right direction, and hopefully there will be more to report as time unfolds.

* Actually there is a concept inventory for calculus. EDIT: I had a comment here originally about the CCI being hard to get hold of, but it appears (see comments) there are good reasons for that.

Image: http://www.flickr.com/photos/npobre/

To focus on technique is like cramming your way through school. You sometimes get by, perhaps even get good grades, but if you don’t pay the price day in and day out, you never achieve true mastery of the subjects you study or develop an educated mind.

Did you ever consider how ridiculous it would be to try to cram on a farm—to forget to plant in the spring, play all summer and then cram in the fall to bring in the harvest? The farm is a natural system. The price must be paid and the process followed. You always reap what you sow; there is no shortcut. This principle is also true, ultimately, in human behavior, in human relationships. They, too, are natural systems based on the law of the harvest. In the short run, in an artificial social system such as school, you may be able to get by if you learn how to manipulate the man-made rules, to “play the game.” In most one-shot or short-lived human interactions, you can use the Personality Ethic to get by and to make favorable impressions through charm and skill and pretending to be interested in other people’s hobbies. You can pick up quick, easy techniques that may work in short-term situations. But secondary traits alone have no permanent worth in long-term relationships. Eventually, if there isn’t deep integrity and fundamental character strength, the challenges of life will cause true motives to surface and human relationship failure will replace short-term success.

Covey, Stephen R. (2009-12-02). The 7 Habits of Highly Effective People (pp. 21-22). RosettaBooks. Kindle Edition

It seems like this quote is pretty dense in implications for educational systems, student approaches to learning, faculty approaches to teaching… what do you think?

This makes a lot of sense in the health sciences because as a practitioner, theory and raw information are important, but it’s the practice itself that really matters. (Just ask my wife, who is now successfully sticking people with needles on a regular basis.) If I were sick, I would certainly feel a lot more comfortable having a nurse who’d spent his or her class time practicing being a nurse with the active guidance of an expert nurse than I would someone who’d sat through a bunch of lectures by a nurse, even if they were great lectures. As my colleague put it, nurses have to be out-of-the-box thinkers who can adapt quickly to rapidly-changing and complex situations. I’d add that the best nurses do this with unfailing professionalism and empathy. Can you possibly learn this in a lecture?

As obvious as this seems, what my colleague mentioned next was interesting too: they are getting from some students the same kind of resistance I received when teaching the inverted MATLAB course. Many students simply want to be lectured to. When I taught the MATLAB course inverted, all of the students were initially uncomfortable with the course design, some vocally so. I invited them to come in individually to discuss it, and I asked them: What would you gain from regular in-class lectures that you are not getting from the screencasts and lab activities now? Unfailingly, the objections ultimately boiled down to: I just feel better when lectured to, even when they themselves could point out the educational advantages of a more active class. My colleague readily agreed the same thing was going on with that (small!) subset of nursing students.

My MATLAB students were fundamentally good students, and so are all of our nursing students now, so this isn’t about “good” or “bad” students. What I think this illustrates is that there is a cultural expectation about how college classes ought to go that is very hard to change. Many students — and faculty! — in higher education are sold on what I called the renters’ model, which is basically transactional. I pay my money and inhabit this space while you take care of my needs, and when I’m done I’ll move on. The inverted classroom is one style of teaching that insists on ownership. There will be some friction when two fundamental conceptions of class time are in such disagreement with each other, no matter how much sense it might make in your content area.

For those who have encountered this kind of resistance, whether it’s from the inverted classroom or some other style of course design and teaching, how did it go for you and your students?

Image: http://www.flickr.com/photos/chrysti/

Here’s why:

• At about 2:00 PM local time in Nanchang, China on January 27 — a Sunday — Penny was brought into the room where my wife and mother-in-law were waiting, and they met for the first time. Lots of pictures were taken with our Canon PowerShot digital camera. This was 2:00 AM local time in Indiana.
• A couple of hours later, the three of them were back in their hotel room. My wife plugged the camera into the laptop we’d borrowed from my college and downloaded the photos.
• She then proceeded to go to Walgreens.com (of all places), upload the photos, and request print copies to be picked up at the Walgreens that was on the way from our house to church. She also sent them to me through email.
• At 6:00 AM Indiana time (6:00 PM China), I woke up to find the photos — just a few hours old — waiting for me in the inbox. In return, I made a quick video recording of myself saying hello to my newest daughter, so she could learn my face and voice before they all came home, and sent that to China.
• At 7:30 AM Indiana time, the older daughter and I left for church and swung by Walgreens on the way to pick up the print photos that my wife had sent in from the hotel room in China.
• At 8:30 AM, we were passing those photos around to our church family.

So within six hours of receiving this beautiful child to our family, we had snapshots plastered all over the walls of our house and our friends’ houses and were swapping photos and video back and forth across twelve time zones. It was as if Penny were already home.

It was at that moment that I realized what a force for good technology can be in the lives of human beings, and it was the moment I realized I wanted to commit myself to understanding ways technology can be used to make life, and learning, more meaningful. We can’t accept technology uncritically, either personally or in education, but we also can’t ignore the power with which it can enable us to engage with our own humanity and become more fully human to ourselves and each other.

Image: Photo I took at the Indianapolis airport when my two daughters first met each other.

When I looked out the door and saw the image you see in the photo, I naturally grabbed the snow shovel, walked out the door, and started clearing off the walkway and the van. I got some curious looks from my neighbors, as if to say: What are you doing? We are paying rent not to have to do stuff like this. And it’s true: The apartment manager usually comes through shortly after a snowfall and clears off the walkways. Usually. But who knows? Maybe he won’t come today. And anyhow, even though I don’t technically own the apartment, I do have a sense of ownership about it, and it just seems the right thing to clear off not only my walkway but also my neighbor’s. Some of my neighbors, on the other hand, take the renters‘ approach and prefer to let the guy they’re paying do the job — whether or not it actually gets done.

The difference between an ownership and a rental mindset is one that we educators encounter all the time with our students. Students engage in one kind of mindset or the other in our classes. The rental mindset says, I am paying the rent, and as long as I pay, I expect the management to take care of my needs. The ownership mindset says, on the other hand, I am invested in this, and although some things are not my responsibility (like plowing the city streets or running the fire department), I choose to take responsibility for myself because it matters to me.

We want students to own, not rent, their education. Ultimately, which kind of mindset students adopt is a choice that only they can make. But while we can’t make students take ownership (just like you can’t make people move out of an apartment into a house), we can make the decision to choose ownership easier or harder through the choices we, as instructors, make when we design classes and learning experiences.

For example, rather than dictating what content students learn in a course, we can instead design courses around clearly-stated learning objectives and give students some latitude as to how they will show us they’ve mastered those objectives. For instance, in the computer workshop in Calculus 3 this past week, I wanted students to use Mathematica to investigate how parameter values affect the behavior of a curve in 3-space. I could have done this by giving students a single vector function with two parameters to study. But instead, I gave three options and had each working group choose one. By giving just a little bit of free choice, students gained a little bit of a stake in the process and thus a little bit of ownership. The object of the course is not to “cover material” — it’s to meet learning objectives, and by letting students choose how they want to do this, they are shoveling their own walkways in an educational sense.

We can also encourage ownership by moving away from instructional designs and methods that promote dependency on the “manager” for “services”. I’m thinking primarily of lecture. Lecture has its uses in certain cases, but it’s clear that a lecture-based course wants the learner-instructor relationship to be primarily one-way. The instructor is paid to produce, and the students consume — it’s a renter’s paradise. But it’s an unsustainable practice for students, who will all be moving into positions in life where they are supposed to be the producers. Some students come into a lecture course with an ownership mindset — like I came into my apartment — and so they naturally take ownership of their learning. But if the goal is to get all students incrementally closer to ownership, a model that disincentivizes ownership is not going to succeed. However, if we choose instructional models that promote student responsibility — like the inverted classroom, peer instruction, or project-based learning — even in small amounts, then we are moving in the right direction.

What are your thoughts on this? How can we encourage ownership rather than rental in the way we design and handle our classes?

Image: I took that photo from the front door of the apartment on 1/29/2012. The bluish mound covered in snow is our minivan.

What was it about that show that I found so engaging? For me, the main thing was the deep humanity of the characters. In the first few episodes, it was very easy to pigeonhole them all. Sawyer was the criminal you had to watch out for; Hurley the lovable loser; Kate the fragile beauty. Then, in later episodes, LOST pulled out one of its signature storytelling devices: the flashback, in which the backstory of these characters is revealed piece by piece over the course of several seasons. And you learned that pretty much all of your initial judgments about these characters were totally wrong. The backstory mattered in terms of making sense of the human being before you.

In 2011, I learned that my students and my colleagues — and I, too — are human beings with backstories. It’s almost damning in some ways to say that I “learned” this in 2011. I guess I already had some idea that this was the case. But in 2011, especially upon arriving at my new position and being keenly aware of my own backstory, I started really paying attention. You learn some pretty amazing things when that happens, for example:

• One student showed up for class intermittently and was almost never engaged with anything. Turns out the student was suffering from chronic walking pneumonia and had no health insurance. I got the student connected with a free church-related clinic, the student got some medicine, and the student proceeded to be OK in the class.
• Another student missed class quite a bit and almost never turned in homework. Turns out that the student was wrestling with huge personal and family issues, had a child to raise, and was in school in the first place because the student’s former job had gone under. That job involved construction, and it turns out the student was highly proficient in remodeling houses and had a really keen sense of geometry and kinesthetically-oriented ideas.
• Yet another student approached me after class with a question, and while talking to the student, it turns out that the student had already finished a degree in finance some years before, had spent a year or three in Chicago in a high-powered analytical trading environment helping to write database code, didn’t care for the job situation, quit, and now was back in school — without loans, since the student already had another degree — getting a degree in computer science to do the coding thing full-time.

You’d be crazy to think these backstories don’t matter. It would be nothing to write off the first student as “weak” or “disinterested” or “unprepared” (the euphemism we math teachers often use to dismiss students who don’t meet our expectations), but actually none of those were true — the student was just sick. The second student could be similarly written off, but in fact the student’s backstory not only shows the student was capable of hard work, it also gave me an avenue for teaching the student more effectively (i.e., use lots of kinesthetic examples and references to building things). When I was answering the third student’s question — about conditional statements — the student’s backstory put me in mind to describe conditional statements as database queries, whereupon the student instantly understood what was happening.

So, I learned that every student has a backstory, and part of my job is to learn that backstory — possibly piece by piece over the course of the semester — and put it to use. It helps me not only to teach students but also helps me to remember that students are not objects: They are complex, evolving human beings undertaking an immensely complicated task in pursuing higher education, and they deserve my respect for that.

Don’t write students off! There is always a backstory, and it matters.