in triplicate

Academic grades, as conventionally understood and used, have three distinct audiences, and they serve a different function for each audience. Grades are problematic largely due to the ways these functions are at odds with each other.

The three audiences are:

• The student. In the context of a single course, assessment and feedback are necessary parts of the educational process. They let the student know how they’re doing as they go along. Traditional letter grades are a crude form of feedback, and where they are used students rarely have a chance to meaningfully respond, other than to try to adapt for the next assessment. Alternative grading systems often treat the entire course duration as a traning period, in which the student can respond to feedback through revision, reassessment, or other forms of demonstrated proficiency following an initial evaluation.
• The institution. Let’s be as generous as possible and assume that the primary goal of the college or university in which a student is enrolled is to educate that student in such a way that they reach their fullest potential. This requires communication among the numerous instructors that student will encounter throughout their studies, and also coordination with the other offices and institutional bodies that are present to support the student. A grade can be a succinct summary to the next instructor in a sequence about how developed are the student’s prerequisite skills. It can also be a signal to the institution about how well the student is progressing in their overall academic career. (This is the reason for “mid-semester” grades that can be used to alert the school when a student needs assistance or intervention.)
• The outside world. Outside of the institution (and inside as well, to an extent), grades become currency that the student can trade for prestige or opportunities. This currency may be in the form of a GPA (to which all of the individual course grades contribute, despite being largely incommensurable with each other) or, for a more granular view, a transcript (which provides the opportunity to craft a narrative about the grades). Of course, not all persons or organizations outside academia value this currency in the same way. But “good grades” provide an inexhaustible supply of recommendations, and “bad grades” are a perpetual obstacle to be minimized or navigated around.

Thus grades are expected to operate at three different social scales, and also at three different time scales. The feedback to a student within a course is short-term, the communcation with the institution is medium-term, and the message to the outside world is long-term (or as they say, permanent). Much of the “objectivity theater” surrounding the assignment of grades is based on the pretence that these three purposes can be fulfilled by a single summative object.

The fact that the third use of grades has both the largest audience and the longest-lasting effects means that it becomes their dominant purpose, their telos. Student anxiety about grades, for the most part, is caused not by an intrinsic dislike of getting feedback about how they can improve their understanding and performance in a subject, but by the belief that in the end what has the greatest practical impact is the final letter or number they can show to the outside world when the course is done. Institutional concerns about “rigor” are based not on the needs of the student, but the needs of the school to present the final scores to the outside world as meaningful indicators of their students’ quality.

Those of us engaged in the practice of developing and implementing alternative grading systems are primarily focused on the first and smallest-scale purpose of grades, providing a useful feedback process to the student. Yet our systems must also interface with the institutional and outside world audiences. At those interfaces lie, in my view, the most difficult ethical questions of grading: how do we support students beyond our time as their instructor? how do we provide an honest evaluation that meets the needs of all three audiences? to whom are we primarily responsible? is the merging of the three functions into a single metric flawed in such a way that it needs to be overthrown?

Honestly, I think (at the time of this writing) that grades are most useful at the institutional level. If it were not for the outward-facing use of grades, they could serve as a quick, qualitative (not quantitative) shorthand in communicating among the internal parts of a college or university what are the needs or successes of an individual student. (To be supplemented by more personalized detail as necessary.) Within a course, as we’ve seen, any number of assessment/feedback systems can work, as long as they’re built on clear communication and building trust in the student-faculty relationship. As for the presentation of grades to the outside world, well, that’s where the dirty work happens.

using calculus to understand the world

In my last post, I wrote about how I returned to teaching related rates in my calculus class and ranted a bit about the inanity of most related rates problems. There I mainly discussed the difficulty in reading the statement of such problems and how to make the questions they raise seem more natural. I’d like to expand on this theme with some more examples.

One feature of mathematics that doesn’t get emphasized enough, IMHO, is that it is a science, and as such is based in observation. Often, either we lead students through abstract reasoning to a previously unanticipated result, or we prove things that are so self-evident that the notion they need proof is itself baffling. Now, in the world of professional mathematics, it is true that even apparently obvious facts need proving (remember that “to prove” just means “to test”), and we often do get excited when we are led to something unexpected and beautiful. That is because we have learned how to use and trust our logical skills to examine the truth of something, and we delight in the uncovering of new truth by means of those skills. But even when a result is surprising to an audience, and even if it was at first surprising to the speaker, it is no longer so. A mathematical researcher plays with ideas until she notices something interesting, and then she tries to understand why it is so. That’s the exciting part of math, and that is what I believe we can share with our students through the process of modeling.

My goal in teaching related rates has become to ground as many questions as possible in direct observation. When I ask about the sliding ladder, as I described in my last post, before setting up the math but after asking students what they think will happen, I demonstrate by leaning a ruler against a book and slowly pulling the bottom end away. What one notices in this experiment is that the top end of the ruler moves very slowly at first, and very quickly just before reaching the ground. The speed in the final moment is so great that one is tempted to think the person pulling the bottom end lost control, and gravity took over. (This is even more credible when using the much larger, heavier ladder in a demonstration.) But the math shows that even if the person keeps complete control and moves at precisely the same speed, the same effect will occur. Let’s see why.

The exact length of the ladder doesn’t matter, of course, so call it $L$. If $x$ measures the distance from the wall to the bottom end of the ladder and $y$ measures the distance from the floor to the top of the ladder, then we have $x^2 + y^2 = L^2$. Then we differentiate both sides with respect to time and get $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$, or \[ \frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt}. \] At this point most related rates problems would ask you about the size of $dy/dt$ for some particular values of $x$, $y$, and $dx/dt$, but look at how much we can determine just from this related rates equation: when $y$ is larger than $x$, the top end is moving more slowly than the bottom end, and conversely when $x$ is greater than $y$, the top end is moving more quickly than the bottom end. There is just one moment when the two ends are moving at the same speed, which is when $y = x$, or in other words, when the ladder is at a 45 degree angle. And as the distance between the top end and the floor approaches zero, the speed of the top end approaches infinity. (Not physically possible, of course, but it explains why there’s such a quick movement at the end of the process.) There; now I feel like I’ve learned something!

I have five more examples to illustrate how much more interesting I think related rates are when tied to direct observation. This will probably belabor the point, but unfortunately these examples are also stripped of any interest by focusing too much on a single moment in time, which is what every standard textbook does with them.

The next example involves inflating a balloon. This, again, is easy to demonstrate. I can’t take a deep enough breath to fill the whole balloon at once, but even with two puffs, exhaled at a near-constant rate, it’s obvious to students that the size (i.e., diameter, or radius) of the balloon grows more quickly at first, then more slowly. Anyone who’s worked with an air or helium tank has surely experienced this phenomenon. Why is this happening? And how much more slowly is the diameter increasing as time goes on? Here there’s very little modeling involved; essentially the entire model is provided by assuming the balloon is a sphere and using the formula for the volume of a sphere in terms of its radius, $V = \frac{4}{3}\pi r^3$. Differentiating with respect to time gives the relation $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$, so \[ \frac{dr}{dt} = \frac{1}{4\pi r^2}\frac{dV}{dt}. \] Some books reach this equation, but as with the ladder they again jump to plugging in values for a specific time, rather than noting the following: if $dV/dt$ is constant, then $dr/dt$ is inversely proportional to the square of the radius! And even more, $4\pi r^2$ is the surface area of a sphere with radius $r$, so this relationship between rates is directly related to the fact that the derivative of the volume of a sphere with respect to its radius is the surface area! That is, the size of the surface of the balloon is what determines, together with the rate the volume is increasing, how quickly the radius is increasing.

A similar phenomenon happens with the standard filling-an-inverted-cone problem. The demonstration here involves a martini glass and some colored water. (As I promise my students when doing this experiment, it’s just water.) My martini glass is about 12 cm across on top, and about 8 cm deep, giving it a volume of 300 milliliters. (That’s about 10 ounces, the size of two regular martinis—you don’t want to fill this glass with gin and drink it too quickly.) Having a nice big glass is useful for this demonstration: if I pour at a constant rate, the water level rises much more slowly near the top than near the bottom. The math shows just how much more slowly. The volume of a cone with height $h$ and base radius $r$ is $V = \frac{1}{3}\pi r^2 h$. From the geometry of this situation (using similar triangles, for instance), for my glass the radius of the surface of the water is always three-quarters of the water’s depth (here interpreted as height). We could use the relation $r = \frac{3}{4}h$ and substitute into the volume formula to get rid of the variable $r$, but there’s also no harm (as my students taught me) in differentiating first, using the product rule: \[ \frac{dV}{dt} = \frac{\pi}{3} \left( 2rh\frac{dr}{dt} + r^2\frac{dh}{dt}\right). \] Notice that this formula is valid for all cones varying in height, radius, and volume, whether or not the height and radius are linearly related at all times. The most obvious quantity of interest (assuming constant $dV/dt$) is $dh/dt$. From $r = \frac{3}{4}h$ we get $\frac{dr}{dt} = \frac{3}{4}\frac{dh}{dt}$, and also $h = \frac{4}{3}r$. The reason to solve for both of these quantities is that, by keeping both $dh/dt$ and $r$ in the equation and substituting out $h$ and $dr/dt$, we get $\frac{dV}{dt} = \frac{\pi}{3}\left(2r^2\frac{dh}{dt}+r^2\frac{dh}{dt}\right)$, or, after solving for $dh/dt$, \[ \frac{dh}{dt} = \frac{1}{\pi r^2} \frac{dV}{dt}. \] First of all, the ratio between the height and the radius has disappeared, so this formula now works for any inverted cone, not just my martini glass. And second of all, just as with the balloon, the rate at which the height increases depends on the “surface area” that is expanding, which in this case is just the base of the cone! Thus, again, the reason the water level rises more slowly near the top of the glass has a clear geometric interpretation. (Here’s a real-world application: I argue this works to the benefit of bartenders, who can pour into a martini glass fairly quickly without risk of overflowing, because the beverage level rises slowly near the top of the glass.)

I took the next two examples from Cornell’s Good Questions Project; come to think of it, it may be these questions that first planted in my head the idea of looking at related rates problems over time, without numbers. The situations are again standard for related rates problems, but the conclusions are much more interesting than a single rate at a single moment.

Consider an actor (say, Benedict) on a stage, illuminated by a light at the foot of the stage. Benedict casts a shadow on the back wall; how does the length of his shadow vary if he walks towards the light at a constant speed? The demonstration of this situation is particularly exciting, because you get to turn off the classroom lights, pull out a flashlight and a doll or figurine, and watch what happens to the shadow of the doll/figurine/actor on the wall as it moves towards the flashlight. Students observe that at first the shadow grows slowly (when the figure is close to the wall), then more quickly as he approaches the light. Modeling this situation generally provides the first major geometric hurdle for my students, because it involves the imagined line that emanates from the light, passes by Benedict’s head, and finally reaches the back wall, thereby determining the height of the shadow. (I wonder if many of them have never thought about the geometry of how shadows relate to the objects that cast them.) I’ll let the reader work out the fact that, if Benedict’s height is $h$, the distance from the light to the back wall is $D$, the distance from Benedict to the light is $x$, and the height of the shadow is $s$, then $\frac{s}{D} = \frac{h}{x}$. (Hint: use similar triangles.) Here the only variables are $x$ and $s$, so the related rates equation is \[ \frac{ds}{dt} = -\frac{hD}{x^2}\frac{dx}{dt}. \] Students are at first perplexed by the negative sign: shouldn’t the shadow be increasing? If so, why does its derivative appear to be negative? Then they realize: ah, if Benedict is walking towards the light, then $dx/dt$ is negative, so $ds/dt$ is in fact positive! And so it becomes clear that the height of the shadow increases much more rapidly when Benedict is near the light than when he is near the wall. (I generally give specific values for the height of the actor and the distance from the wall to the light in this question, so that it’s more obvious which values are constant.)

I don’t have a standard demonstration for this next problem, because I use it as a quiz question (although maybe not anymore, now that I’ve written about it here), but it’s easy enough to devise an experiment. This situation is similar enough to the previous one that its result is a bit surprising. Suppose a streetlight at height $L$ is the only source of illumination nearby, and a woman (say, Agatha) of height $h$ walks at a constant speed away from the light. As she gets farther away from the light, does her shadow grow more quickly, more slowly, or does it grow at a constant rate? If $x$ again denotes the distance to the light (well, really from Agatha’s feet to the base of the lamp, which is not the same as her distance to the source of illumination), and $s$ is the length of Agatha’s shadow, then similar triangles produce the relation $\frac{s}{h} = \frac{s + x}{L}$. We can rearrange this into a simple proportion between $s$ and $x$: $s = \frac{h}{L-h} x$. (Here’s an interesting feature of this equation already: it only makes sense if $h < L$, that is, if Agatha is shorter than the lamppost!) Now we differentiate to get \[ \frac{ds}{dt} = \frac{h}{L - h} \frac{dx}{dt}. \] So if Agatha’s speed is constant, then her shadow’s length is also increasing at a constant rate. This example shows especially well why it’s dumb to look at related rates at a single moment in time. Most book exercises of this sort ask how quickly the shadow is growing when Agatha is at a particular distance from the lamp. But it doesn’t matter how far away she is, and the math proves that it doesn’t matter.

There’s a risk in related rates exercises to always resort to problems that only involve differentiating polynomials, so here’s an example that uses trigonometric functions. The demonstration I use: I walk back in forth in front of the class and tell the students to be mindful of what their heads do as they follow my movement. After a couple of times, several of them observe that their heads must turn more quickly when I’m closer to them. I point out that this is something anyone who’s had to run a video camera at a race must be aware of. (It’s also apparent to someone riding in the passenger seat of a car, keeping their gaze fixed on a single tree or other immobile object: for a long time, your head turns little, but when you’re close to the object, you have to turn quickly to keep it in view.) I generally set up the problem on the board as though it is taking place at a racetrack. Suppose a runner is moving along a track (let’s assume it’s straight for simplicity) at $v$ feet per second. You’re watching from a position $D$ feet away from the track. How quickly does your head need to turn to keep following the runner? The answer depends on how far away the runner is. One has to introduce a reasonable coordinate system and some useful variables: good choices are the position $x$ of the runner relative to the point of the track closest to you, and the angle $\theta$ by which your head is turned from looking at this closest point. Then we get the relation $\tan\theta = \frac{x}{D}$, and differentiating with respect to time results in the equation $\sec^2\theta \frac{d\theta}{dt} = \frac{1}{D} \frac{dx}{dt}$, or \[ \frac{d\theta}{dt} = \frac{v}{D} \cos^2\theta \] (using the assumption that $dx/dt = v$). When $\theta = 0$, so that the runner is closest to you, the rate at which your head turns is $v/D$, which depends only on how fast the runner is going and how far away from the track you are. (Notice that the units work out: the radian measure of an angle is technically dimensionless, and so we expect its rate of change not to have any dimension other than 1/time. Since $v$ has dimension of distance/time and $D$ has the dimension of distance, $v/D$ has the dimension 1/time.) As $\theta$ increases (in this scenario, $\theta$ is never greater than a right angle), the change in the angle of your head to follow the runner happens more slowly, because $\cos^2\theta$ is closer to zero.

These are just a few examples of standard situations involving related rates that become much more interesting when the myopic attention to a single moment in time is removed. I’m sure most readers of this post can do the calculations I’ve shown on their own, but the tendency to hone in on a single rate at a single point in time is so entrenched that I wanted to show how much more interesting related rates become when that element is removed. I don’t know that my students are better at solving related rates problems than other students, but I have noticed that they’re much less likely to insert specific quantities into a relation before it’s necessary than when I taught the subject years ago. I haven’t had time to strip all such problems of the detritus that comes with wanting a numeric answer, but I believe our understanding (and our calculus students’ understanding) of the world will be much improved by making the effort to transform these problems into meaningful questions.

Here are two other examples that I won’t work out in detail. One scenario has a boat being pulled into a dock by a rope attached to a pulley elevated some distance above the boat. If the rope is pulled at a constant rate, the boat in fact speeds up as it approaches the dock! (I tried demonstrating this once with a string tied to a stuffed animal pulled across a desk, with moderate success.) Another common type of problem considers two boats moving in perpendicular directions (or cars moving along perpendicular roads), and asks at a certain point in time whether the distance between them is increasing or decreasing. That’s silly. Why not establish the relation between them, and ask at what times the distance is increasing, and at what times the distance is decreasing? If there’s a time when the rate of change in distance is zero, then the boats (or cars) are at their closest (or farthest) positions, which connects to the study of optimization, which has its own set of issues…

P.S. I should have known better than to look at Khan Academy’s treatment of related rates. His videos show all the marks of what is classically wrong with these problems: the irrelevant information of what variables equal at a single moment in time is presented up front along with everything that’s constant in the situation, and in the end the answer is a single, uninformative number. Even when an interesting equation is present on the screen, Khan rushes past it to get to the final number. How can we get our students to ask and answer more interesting questions than these, about the same situations?

thinking about thinking about thinking

Last year, I learned the word “metacognition”—roughly, “thinking about thinking”—and its usefulness in pedagogy. Put simply, both students and instructors can better direct the learning process when they are aware not only of what students understand about the course material, but of what the students think they understand. In the fall, I saw my students doing this almost automatically as they wrote brief essays over the course of the semester; many chose to write about their own understanding of mathematics (specifically calculus), what doing math means, and how math is used in the wider world. (More on these essays later.)

This spring, I want to make the metacognitive aspect more explicit in my classes. I will be teaching a multivariable calculus class and an “advanced calculus” class (which will be focused on the geometry, topology, and calculus of manifolds in Euclidean space of arbitrary dimension). Each class will have a distinguishing feature that lends itself to self-reflection. In multivariable calculus, I will be using standards-based grading (SBG) in place of traditional number-and-letter grades, in order to better focus students’ efforts towards improving specific skills, rather than a single, not very helpful grade. In advanced calculus, students will complete biweekly projects that tie together ideas from lectures and prod them to explore new ideas on their own.

How could I use these to encourage “thinking about thinking”? In the first case (multivariable calc), I’ve been considering also having brief, weekly self-reflections in which students will describe what they think the main ideas for the week were, and what they found confusing. (I’ve borrowed this idea from some other source, which I can’t find at the moment.) It seems like it would be easy enough to put the relevant standards on the self-reflection form and have them mark where they think they stand. In the second case (advanced calc), I am trying to follow this principle in creating projects:

It’s OK for new ideas to be abstract, as long as they’re familiar in some way. And it’s OK for them to be unfamiliar, as long as they’re concrete. If I, as a teacher, try to introduce an idea that is both abstract and unfamiliar, then it is likely most students will be lost.

(Paraphrased from a comment made by Steven Strogatz about the expository essays he has written for NYTimes.) I plan on stating this principle outright to my students: when you start a project, you should expect to find ideas that are either familiar or concrete. Look for those, and build your understanding around them. Even better, take that principle and use it to transform your approach to learning. If an idea is unfamiliar, find a way to make it concrete (often by looking at specific examples). If an idea is abstract, connect with something familiar.

I think these two ways of dealing with metacognition are appropriate for the different levels of the classes. Multivariable calculus remains among the set of math classes that are not always taken for the sake of math itself; often students approach it with the paradigm of “mathematics as tool”, and so their reflection on their understanding itself needs a concrete form and guidance. This advanced calculus course, on the other hand, is a transitional course into higher areas of mathematics and is more likely to have students who are interested in math for its own sake; these students should be practicing independent learning and encouraged to make self-reflection a part of that process. These ideas are relatively recently formed in my own head, and I plan to fill them out more over the next two weeks as I prepare for the start of the semester.

One final credit: I first encountered the idea of “metacognition” during a presentation by a representative of McGraw–Hill about one of their online products. While I have not had a chance to explore this product in depth, I am grateful to the presenter for sharing this idea.

whiteboard comparison: Educreations vs. Doceri

Update (3/28/2015): This is still one of my most-read blog posts. Not that I mind the traffic, but I feel I should point out that this information is probably seriously out of date. I haven’t used either of the apps described here in over two years—not because I grew to dislike them both, but because the project for which I was using them got put on the back burner. And how much have iPads and iPad apps changed over the last two years? Which is to say, if you’re reading this in 2015 or later, you shouldn’t accept my take as definitive, but try these apps (or their other competitors) for yourself and decide what features you like.

Update (9/24): It seems that enough people are still finding their way to this post that I should correct some of the misconceptions I had when I wrote it. Where this happens, I’ll strikethrough the original text and add what I’ve learned. Two of the big ones are corrected in the comment left at the bottom by the Doceri team, but I figured no use making people work for information.

As soon as I got my iPad (as a gift) last spring, I wanted to see how I could use it for education—not just for classes, but also to help produce interesting materials for general consumption. One type of program I quickly came across was “whiteboard apps”, which can be used to produce videos that include both an audio component and a screencast of a faux-whiteboard. Thinking this was my chance to become just like C. G. P. Grey and Vi Hart, I downloaded a couple and started playing around. Recently I started using another one. Here I’m going to compare the two I’ve tried the most, the Educreations Interactive Whiteboard produced by Educreations, Inc., and Doceri, produced by SP Controls, Inc.

Let’s start with what they both have:

• both have responsive marker tools, with a variety of colors;
• both allow you to import outside images and incorporate them into presentations;
• both allow you to set up slides before recording;
• both allow you to pause during recording;
• both allow you to email the results, or present them on the web.

In short, both are very useful for making short, expository videos for classroom or other use.

Here are the specific pros for Educreations:

• The interface is very intuitive. There’s a minimum of buttons to accomplish the task at hand, and it’s clear basically from the start what every one does. (Doceri’s interface took me a while to sort out.)
• The marker’s appearance is very polished; during writing, the app looks and responds to pressure as much like a real marker as one could imagine. (The marker in Doceri is a fixed width throughout every stroke; so, while the program follows the movements of handwriting to produce text, it doesn’t really emulate handwriting. I actually correct this below. There are lots of options with Doceri, including fixed-width and pen-like.)
• Images, once imported, can be treated as objects—moved around, scaled, and even rotated during the course of a video. (Once an image is imported into Doceri, it becomes a fixed part of the background. This one I goofed because I tried to select images in Doceri with the lasso tool, the way other objects are selected. But press-and-hold on an image will select it for you.) There’s also support for adding typed text if handwritten isn’t the effect you want for parts of your video. (Doceri doesn’t have this feature.)
• Once you save a video, it’s immediately available online for viewing on the Educreations website.

Here are the corresponding pros for Doceri:

• Lots of flexibility. More on this shortly, but one has a lot of control over features such as when objects and writing appear on the slide and how quickly they appear.
• The width of the marker can be specified, as can the level of opacity and the amount of spacing between individual dots in a stroke (in case you want to draw dotted lines without lifting between every dot). You can choose the color from a palette or a color wheel. There are also multiple options for the type of head, such as pen, felt-tip, paintbrush, and highlighter. (Actually, I just played around with this some more, and my above assessment of the marker as merely fixed-width wasn’t fair. Sorry about that.)
• In addition to the variety of writing tools, there are several choices of background besides just plain white. It also has construction tools for producing straight lines, rectangles, ellipses, and arrows. Objects created inside the app—including writing—can be moved, scaled, rotated, and copied.
• Videos can be uploaded to YouTube or Facebook from inside the app. (Educreations appears to have a proprietary format, so that videos can only be viewed on their website, although they can be embedded elsewhere on the web.)

But here is the biggest difference, the one that for my purposes is the make-or-break feature: in Doceri, slides and recordings are stored as separate projects. You can lay out all of the materials in order, set up every stroke of every slide, establish pause points, and then use them as the basis for your recording, or even multiple recordings. But you don’t actually have to make a recording at all. In fact, the slides can be used for PowerPoint-style presentations by connecting to a computer remotely. (I haven’t tried this feature, but I’ve seen a lot of reviewers say they like it.)

In Educreations, you can still set up slides ahead of time, but if you mess up during the recording, you have to start over completely and set it all up again. (This was the aspect that caused me to start looking for another resource.) The visual and audial components are intrinsically linked, and until you have made a completed recording, you cannot save the project and set it aside to work on another without trashing the whole thing.

One shortcoming both apps share is that, although you can pause during recording, you basically have to make an entire video in one take. There’s no option to splice different recordings together, or to replace or remove a particular portion of a recording. But given that both apps are free, I’ll just take this as a sign that I should buckle down and learn how to use iMovie. Here Doceri wins again. Not only can you record multiple times using the same set of slides, you can merge recordings (by dragging the icon of one on top of the other), and so you can create your video in “scenes”—no more starting over on a whole five-minute video because the end went awry!

I could say more, but I’ll just finish by giving you a chance to compare the results of my (inexpert) fiddling with these apps. At the beginning of the summer, I made a video about a particular kind of dynamical system using Educreations: you can view it by clicking on the link. At the time I made this video, I didn't have a stylus, so please take that into account when considering the handwriting.

To get some practice with Doceri, I remade the same video today:

You’ll notice that Doceri adds a watermark to the corner of the video. An in-app purchase ($4.99) will remove the watermark—a clever way, I think, for the developers to advertise and get funding. (If you're reading this on August 31, then the purchase is on sale for $0.99.)

I will probably keep using both apps, but to continue my series on dynamical systems, I’m going to stick with Doceri.

Update: The Doceri team has left a comment below explaining some of the features I hadn't discovered!

Update: The Doceri team seems very happy to have received this feedback. On their blog yesterday, they featured a quote from this post. They also gave an explanation for why I didn’t try their app at the beginning of the summer: they just added recording screencasts with the latest update, in July.