Monday, June 01, 2015

What it means to be truly educated

I came across a delightful quote by Noam Chomsky about what it means to be truly educated that I couldn't agree with more:
"... it's not important what we cover in the class, it's important what you discover. 
To be truly educated from this point of view means to be in a position to inquire and to create on the basis of the resources available to you which you've come to appreciate and comprehend. To know where to look, to know how to formulate serious questions, to question a standard doctrine if that's appropriate, to find your own way, to shape the questions that are worth pursuing, and to develop the path to pursue them. That means knowing, understanding many things but also, ... to know where to look, how to look, how to question, how to challenge, how to proceed independently..."

Chomsky's response gets to the deep connection between education and autonomy. What it means to be educated is to have the motivation and the ability, to pursue inquiry and discovery on one's own. The goal of our academic institutions should be to cultivate in each student the confidence, the drive, the tools needed to pursue and discover truths about how our world is, and to articulate and shape how it ought to be. Our classrooms should not be focused only on the transmission of known facts and methods.

Friday, May 01, 2015

Eine Kleine Nacht Integralrechnung (A Little Night Integral Calculus)

I had asked my middle-schooler if he could spare me some time after dinner to talk about some mathematical ideas and he had agreed. So I rushed to put the plates away, and we gathered with papers and a marker. 

After a preliminary conversation on the nature of acceleration in cars, we decided to focus our discussion on the topic: how far does a dropped object fall in a given period of time? I illustrated it by dropping the marker from a small height. (Assuming no air resistance, I added.)

We started with the basic notion of acceleration. When we say an object is experiencing constant acceleration, what do we mean? I took up the example of the constant acceleration due to gravity, which we call g. What is the value of it, I asked him? 9.8 he answered, remembering this constant from our previous discussion. 

We talked about the units of acceleration, meters per second-squared, which is telling us the additional speed increase every second. So in this case, the speed of the falling object increases every second by 9.8 m/s, I said to him, and he nodded.

I raised and dropped the marker again. We watched it fall with a thud.  What was the speed at the very moment when it started its fall? I asked. 0, he said. 

How about a second later? 9.8 m/s, he said. 

How about in two seconds? 9.8 + 9.8, which is 19.6 he said. I agreed, right, two times the acceleration g. So then, he asked, to clarify, is the speed 2g's after 2 seconds?

I clarified that we normally refer to g's when talking about acceleration, which in this case will always be 1g, but numerically, yes, the speed at this point, measured in m/s, would be 2 times the value of g. And we extrapolated from there how it would 3 times g after 3 seconds, and so on. 

I wrote down the formula: speed = acceleration x time, which corresponded to our extrapolation. I drew points on a plot with the x-axis labelled time and y-axis labelled speed: (0,0), (1, 9.8), (2, 19.6), (3, 29.4), and the line that goes through those points. We talked briefly about how this fits the form of the equation of a line (y = m.x + c) which he saw recently. 

Now, I asked, how much distance did the object travel in the first second? He said 9.8 meters. Ah, I responded. That would be true if the object was going 9.8 m/s for the entire first second. But in fact it was starting at 0 m/s and accelerated to 9.8 m/s only at the end of that second. So you would expect it was something less than 9.8 meters, no? Ah yes, he said, smacking his head, I forgot!  

Let's see if we can figure it out, I said. I drew a horizontal line from 0 s to 1 s at 9.8 m/s and then a step up to another line from 1 s to 2 s at 19.6 m/s then another line from 2 s to 3 s at 29.4 m/s. Imagine this object, I said. It moves at 9.8 m/s for the first second, then at a constant speed of 19.6 m/s the second second, and then a constant speed of 29.4 m/s for the third second. How far would it travel in these three seconds? We worked it out: 9.8 + 19.6 + 29.4 = 58.8 m.

Then I showed him how the distance traveled in this case could be viewed as the sum of the areas of the corresponding rectangles. I then asked him, do you agree this object is always traveling faster than our free-falling object except at the end of each second when it's equal? He nodded. So this object travels farther than ours. He nodded again.
Now consider another object, I said, which travels 4.9 m/s from 0 s to 0.5 s, 9.8 m/s from 0.5 s to 1 s, and so on. This object's distance traveled would also be the sum of the corresponding rectangles, I pointed out, and would also be higher than the distance traveled by the free-falling object, but a little better approximation. His eyes lit up as I pointed out that we could make better and better approximations, and he could see how the distance traveled by the free-falling object would come to be the area under the line we had drawn showing the speed versus time function for it.

Now, I said, recall that we agreed the speed of the object could be expressed as acceleration x time. He agreed. I continued, the area in question (i.e. under the line) is that of a triangle with this height of "acceleration x time", and a base of length "time". So by the equation for the area of a triangle (half of base x height), we can conclude the total distance traveled would be 1/2 x acceleration x time ^2. 

I pushed on. This is pretty nifty, to have such a clean answer to the question how far does the object fall in a given time. But often we don't get a nice equation like this. Consider the kind of speed versus time that is typical of when I drive the car. It speeds up, stays constant for a while perhaps, then slows down again, maybe stops for a bit at a light, then picks up again, and so on... We get a strange, arbitrary shape. What would be the distance traveled in this case? By analogy to what we just did, it is just the area under this arbitrary curve. He agreed. And how can we figure out this area? I asked him. By drawing those rectangles, he replied... 

His mother, who was listening to us by now as well, chimed in, isn't this Calculus? Yes, I nodded. This is what's called Integration. We talked briefly about Newton and Leibniz and their competing claims to having invented Calculus.  

My son's eyes lit up with a question. He picked up the pen from me and drew a blob on the paper. What about the area of this blob? he asked me. Can you calculate it? I said, sure, using this method of rectangles we could calculate the area approximately. No, but exactly, he asked. Well, not sure I could give you an infinite precision answer, I said. Aha, thought so! he exclaimed, feeling satisfied at encountering a fundamental limit to our understanding of things. 

I mentioned then a couple of other ways to estimate the area of that blob. We could take a picture of a rectangle around it and count how many pixels are inside the blob and how many are outside, then use that ratio and the sides of that rectangle to get the area. We could ask a computer to generate a sequence of random points inside the square and check for each point if it is inside the blob or not, and the average ratio in this case would also converge over large numbers of samples to the right answer (this is called the Monte Carlo method, and was invented by Ulam while working on the Manhattan project). Or we could carve out that shape with uniform depth in some material and measure how much liquid it holds to estimate the area... 

There was a bit of a digression here while we talked about the computability of real numbers briefly and I mentioned Chaitin's number Omega, which is a real number with the property that no algorithm can compute its digits (unlike pi).

My wife wondered if you could not also figure out the area from the perimeter of the blob which would be easy to measure with a thread. Ah, I said, but you can't infer the area from the perimeter. We talked about how a circle and a square having the same perimeter would have different areas. My son agreed though he said he couldn't prove it. I thought we would save that proof for a later conversation, and we ended our discussion for the night...

ps: The title I chose for this article is of course a weak attempt to evoke Mozart's famous piece, which I will never forget listening to at a live performance when I was 19 and everything seemed possible... 

Tuesday, April 07, 2015

Staking a claim

Came across a quote from René Decartes in an article about his mathematics:

"..having determined as I did [in La Géométrie] all that could be achieved in each type of problem and shown the way to do it, I claim that people should not only believe that I have accomplished more than my predecessors but should also be convinced that posterity will never discover anything in this subject which I could not have discovered just as well if I had bothered to look for it (To Mersenne, end of December 1637; AT 1, 478; CSMK, 78–79)."

He seems to me to be staking a claim not only on the innovations that are immediately apparent, but also on all other future discoveries on the subject, that had he only "bothered to look for", he would have made those as well... talk about self-confidence!

Monday, April 06, 2015

Opening the Eye of the Soul

I was talking to someone who noted that though they had done AP calculus in high school, all that math seemed to have been of no use or meaning whatsoever in their life.  Indeed, this person noted, setting aside its role in the narrow fields of science and engineering, is it not true that there is no point to teaching all the math we do in schools, even geometry, to every student given only arithmetic is practically encountered on a daily basis and even that could mostly be done on a calculator?

This is an age old question. A dismissive response to it was said to have been offered by the great Euclid, who, when a student asked what good he would derive from learning Geometry, ordered his assistant to "pay him three obols, for he must profit from what he learns!"

But while Euclid, in this story, treats with contempt the questioner for seeking a material use for Geometry, he doesn't express in clear and positive terms what other good it might bring.

I came across the following quote, from a T.Taylor, in his "Dissertation on the True End of Geometry" (1792), wherein he addresses the most important reason to study geometry more explicitly, rather poetically:

".. if geometry is a speculative science (I mean the geometry of the ancients), it is both desirable for its own sake, and for still higher contemplations, the visions of intellect, to which it is ultimately subservient. For, when studied with this view, it opens the eye of the soul to spectacles of perfect reality and purifies it from the darkness of material oblivion. Away then, ye sordid vulgar, who are perpetually demanding the utility of abstract speculations, and who are impatient to bring down and debase the noblest energies, to the most groveling purposes..."

As it happens, I have been going through a few propositions from the first book of Euclid's Elements with my son in recent weeks. I think I understood why I was doing it only dimly till now...


For anyone that may be interested, here are a few excellent links to study the Elements:

The Elements are quite accessible to be explained to a middle-schooler. There is something really inspiring and meaningful about learning Geometry directly from the Ancient Greeks, from a book first written circa 300 BC. It is still current and makes for excellent training in rigorous thinking and deductive reasoning. It makes it possible to gauge for oneself how brilliant and sophisticated they were, how far along they got with abstract thinking, and helps one see the unbroken threads that connect philosophical and intellectual investigations and the growth of knowledge through the ages.

Thursday, October 23, 2014

On the Virtue of Putting the Cart before the Horse

You may have seen the above video by "DjSadhu" purporting to show how the motion of the planets would appear if we use a frame of reference with respect to which the sun is moving. If not, I'd encourage you to take a look. It's fascinating.

However, Phil Plait has posted an article on pointing out that this viral video is in fact scientifically incorrect, where he says:
It seems right, or looks cool, or appeals to some sense of how things should be. But how things should be and how they are don’t always overlap. The Universe is a pretty cool place, and works using a fairly well-regulated set of rules. We call those rules physics, they’re written in the language of math, and trying to understand all that is science.
My first reaction on seeing Phil's article was, "Yes, that makes sense. As he says, 'not everything cool is science...' "

But then I saw a post by a friend of mine, who was a former faculty member (now working in industry), in response to Phil's article. He wrote that Phil was, well, a "killjoy. Perhaps, [a] scientifically accurate killjoy, but still a killjoy. " His point was that the original video, despite its inaccuracies, served to spark the imagination of many people (as of now, that video has received over 2 million hits!) and that Phil could have offered his corrections in a more constructive spirit rather than proclaiming loudly the incorrectness of the original video.

Reflecting on his comments, I think my friend is really on to something.

Just recently some colleagues and I were having an interesting conversation about why college is boring for many students and we figured out that we college faculty may all be killjoys.

Faculty, due to both nature and nurture the most pedantic of creatures, would rather say something careful and provably correct than show something cool of dubious "truthiness". We would rather build up little by little from abstract fundamentals in the vain (often unfulfilled due to lack of time) hope of reaching eventually concepts one can relate to. We are not inclined to let our students encounter something half-baked that they find amazing and thought-provoking, and then work from that spark of excitement towards the truth.

It is worth pausing to think whether our sneering dismissal of imprecise, careless thinking, our endless harping on rigor, our ceaseless skepticism (all qualities that are essential for our work as researchers, the very qualities we are recognized and lauded for by our peers) might sometimes get in the way of creating the best environment for our students to seek the truth on their own.

Now, let me clear, I am not arguing for teaching students cool-but-unscientific lies or even half-truths. If our own hunger for truth and the desire to share that drive with our students did not ultimately motivate us, we would not be in this line of work. But perhaps there is a place for the incorrect, the absurd, if it is cool enough to draw in the students' curiosity and their imagination, and inspire them to proceed further? It could serve as a starting point for a more careful and rigorous investigation.

Sometimes at least, we should place the cart before the horse (or the sun in front of the planets) because that is unusual and striking, enough to make one stop, stare and think.

Friday, August 29, 2014

The mind is not a bottle

I came across an essay called "De Auditu" ("on listening to lectures"), by Plutarch, the Greek Historian and Essayist, who lived nearly 2000 years ago. It's worth a read, if only to recognize how timeless his words are.

In the essay, Plutarch begins by noting that it is just important to consider and learn how to listen as it is to learn how to offer a discourse. He urges the listener to be patient, to focus on substance not style. He recommends moderation in posing questions, including limiting one's questions to the area of the speaker's expertise. He suggests that even errors in lectures offer opportunities for learning, by motivating introspection into one's own ways of thinking:
Where [the speaker] is successful we must reflect that the success is not due to chance or accident, but to care, diligence, and study, and herein we should try to imitate him in a spirit of admiration and emulation; but where there are mistakes, we should direct our intelligence to these, to determine the reasons and origin of the error. For as Xenophon asserts that good householders derive benefit both from their friends and from their enemies, so in the same way do speakers, not only when they succeed, but also when they fail, render a service to hearers who are alert and attentive. 
He describes an ideal listener's demeanor:
Finally, the following matters, even with speakers who make a complete failure, are, as it were, general and common requirements at every lecture: to sit upright without any lounging or sprawling, to look directly at the speaker, to maintain a pose of active attention, and a sedateness of countenance free from any expression, not merely of arrogance or displeasure, but even of other thoughts and preoccupations. Now in every piece of work, beauty is achieved through the congruence of numerous factors, so to speak, brought into union under the rule of a certain due proportion and harmony, whereas ugliness is ready to spring into being if only a single chance element be omitted or added out of place. And so in the particular case of a lecture, not only frowning, a sour face, a roving glance, twisting the body about, and crossing the legs, are unbecoming, but even nodding, whispering to another, smiling, sleepy yawns, bowing down the head, and all like actions, are culpable and need to be carefully avoided.

He urges listeners to apply themselves and think critically about what they hear:
... let us urge them that, when their intelligence has comprehended the main points, they put the rest together by their own efforts, and use their memory as a guide in thinking for themselves, and, taking the discourse of another as a germ and seed, develop and expand it. For the mind does not require filling like a bottle, but rather, like wood, it only requires kindling to create in it an impulse to think independently and an ardent desire for the truth. Imagine, then, that a man should need to get fire from a neighbour, and, upon finding a big bright fire there, should stay there continually warming himself; just so it is if a man comes to another to share the benefit of a discourse, and does not think it necessary to kindle from it some illumination for himself and some thinking of his own, but, delighting in the discourse, sits enchanted; he gets, as it were, a bright and ruddy glow in the form of opinion imparted to him by what is said, but the mouldiness and darkness of his inner mind he has not dissipated nor banished by the warm glow of philosophy. 
The oft-quoted phrase "For the mind does not require filling like a bottle, but rather, like wood, it only requires kindling" appears here. (It is often mis-attributed to William Butler Yeats in a modified form: "Education is not the filling of a pail but the lighting of a fire.")  With these words, he gets to the very heart of what it means to truly learn.  We can only learn when we are motivated and fully engaged in the learning process. Passive listening is not beneficial.

Modern constructivist education models have sought to reduce the role of lectures in the classroom, to move the role of the teacher from being a "sage on the stage" to a "guide on the side", in part because the art of listening is lost in students who are forced to listen to too many lectures, for years and years. But given that there are still many classes that rely on traditional lectures, and even flipped classrooms require listening to taped lectures outside the classroom, it would behoove students to reflect upon Plutarch's advice and ponder the difference between listening as "bottle-filling" and listening as "fire-kindling".

Saturday, August 16, 2014

Give breaks to improve learning

Came across this article titled "What can the American and British education systems learn from classrooms in the developing world?" 

There are some excellent points here (though perhaps also some idealizations and over-generalizations). One I particularly liked was this: "In the United States, there is the expectation that students are supposed to sit still. You’re told not to fidget and to focus. But scientific research shows that brain activity is significantly heightened after 20 minutes of physical activity. There’s significant value in what you see in the developing world—in between classes, kids run in a field, play in a river, climb a mountain."  

Here is an article about a CDC-led study which describes relevant research findings, including: "Short physical activity breaks of about 5 to 20 minutes in the classroom can improve attention span, classroom behavior and achievement test scores." 

Though sitting still (else being diagnosed as having ADHD) is sadly indeed the norm in most US schools, there are fortunately some schools here where kids are not expected to sit still all day. A number of them fall into the category of "democratic schools" (in which the students have a significant voice in what, where, how to learn).

Older kids and adults do have more stamina, but I do think this applies to the college classroom as well. I'm sure many faculty members have seen their students' eyes glaze over after an hour-long lecture. I give a break or two in my longer classes and find it does help students concentrate on the material better. This fall I may tinker with more frequent micro-breaks, allowing students to stretch and move around a bit more. Active learning techniques including hands-on activities and projects for which students must talk to and work with other students in class also help a great deal with improving engagement and excitement in the classroom and I am thinking hard about these as well as I plan my teaching for the semester coming up. 

Wednesday, August 06, 2014

Academic Contributions

I imagine that nearly all academics ask themselves this question from time to time: "is my work meaningful?"

It occurs to me that one derives meaning from, fundamentally, by making a contribution to others; here are some of the many ways in which an academic's contributions could be evaluated:

  • Contribution to the literature: Has one's work been read and cited by other researchers? How have peers evaluated one's work in terms of novelty, significance, depth, quality? Is there evidence that one's ideas have positively influenced the work of other researchers?
  • Contributions to aid other researchers in their work: Besides papers, has one contributed other materials such as code, tools, data-sets that others could make use of in their research work? Have these been used by others?
  • Contributions to community-building: Human enterprises thrive when we organize into communities. Has one contributed to building a community of researchers? These contributions could be in the form of organizing meetings and workshops and conferences to increase interactions, editorial efforts, organizing centers.
  • Contributions to education: Has one contributed through new courses, surveys, tutorials, books, talks, demonstrations, popular writing, or other materials to educate students, researchers, practitioners and inform the broader public about new developments and ideas? How many have been influenced by these materials and in what ways?
  • Contributions to mentoring: How well has one mentored students?  Has one aided younger colleagues in their professional development? Mentoring is a valuable activity because it enhances the ability of other individuals to make their own effective contributions.
  • Contribution to practice: Has one's work been translated to practice? How has the translation been carried out? What difference has it made in the real world, in the context of that translation? How significant has the practical contribution been?

Wednesday, September 18, 2013

Do what you don't have to

This morning, a student came by during office hours and asked if he could discuss a private matter.  He told me that he was nearly done with all his courses in engineering school, but, to his disappointment, he didn't feel passionate about any of the subjects he had taken. He asked me what was the secret to finding something one has passion for.

I thought about it a bit, aware of the awful weight of responsibility, the need to give a good answer to this bright young man's question, which clearly came from his heart.

I offered him this thought, that I believe passion comes from "doing things no one else has required you to do". I talked with him about how this impulse is in fact thwarted in classrooms with their many assignments, quizzes, exams and canned projects (yes, even mine.) I encouraged him to tinker and do some projects on his own, engage in independent study.

I also told him to try and be mindful of his own inner-voice, to find out for himself what activities he finds rewarding, and seek ways in his life to do more of that. I told him about my own motivations for seeking an academic life --- my realization that I really, really enjoyed teaching my classmates the night before exams what we should have learned in the weeks prior.

Later in the evening, I was still thinking about the conversation. I was thinking to myself how it is that when you feel passionate about something, work or learning doesn't feel like a chore -- it can be exhilarating, as meaningful, as play -- and how the truest form of learning only takes place when you are fully engaged in and enjoying the experience overall. The crux of it is having a clear sense of autonomy --- that you are choosing to do this, you are doing this because you want to, that you could and would stop doing this if it wasn't worthwhile or interesting.

This article by Psychologist Peter Gray makes nearly the same point, in encouraging us all to let our kids play more.

He writes:
The reason why play is such a powerful way to impart social skills is that it is voluntary. Players are always free to quit, and if they are unhappy they will quit. Every player knows that, and so the goal, for every player who wants to keep the game going, is to satisfy his or her own needs and desires while also satisfying those of the other players, so they don’t quit. 
The kind of play he is talking about is not merely light-hearted diversion, though it includes that too, but also the kind of intense immersive play that requires one's full concentration, skill, and forces one to constantly build and improve on one's abilities. Plenty of exposure to this kind of play as a child is indeed essential to self-motivation, creativity, and passionate immersion in learning and work as an adult.

Wednesday, July 17, 2013

Teaching Mathematics as a way of Thinking

I came across today a brilliant blog today about learner-centric mathematics education, called Doing Mathematics.  The writer of this blog, Bryan Meyer, is a high school math teacher in San Diego. He writes that:
Most people think that mathematics is a body of knowledge and/or procedures that must be passed from teacher to student. For many students, this is both irrelevant and unexciting. My belief is that mathematics is a way of thinking and sense making. It is creative, beautiful, individual, and dynamic. It is being curious, asking questions, figuring out why things work, breaking problems apart, seeking regularity, making predictions, and creating logical arguments.  These things we all engage in on a daily basis and have relevance in our lives outside of a mathematics classroom. My goal is to create an educational environment in which mathematical thinking is not only the core of what we do, but runs through ALL of what we do. 
Bryan is inspiring because he does things many teachers wouldn't think of doing, but that strike me upon reflection as a very good thing to do. For instance, in one post, he describes his class working on and discussing a probability problem, at the end of which some students thought one answer was right, while others felt that another answer was right. But he never jumped in to tell them which one was the right answer, even at the end.

This is unnerving to read at first because, of course, the instinct is to say --- wait, shouldn't the teacher announce the correct answer so no one is left with the wrong impression? He chose differently. He writes:
I'm curious what you think and what you would do in this same situation. I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think. 
Bryan's point is that in the long term, showing his authority by presenting the right answer could potentially do harm to their ability and confidence to test out their own ideas. As he writes in a different article,
I am inclined to think that my distaste is not actually for right answers but rather for the students' lack of authority in deciding that answer. As it stands now, students' ways of thinking are always subject to some greater authority (teacher, textbook, video, etc.). As Schoenfeld puts it, students:
"...have little idea, much less confidence, that they can serve as arbiters of mathematical correctness, either individually or collectively. Indeed, for most students, arguments are merely proposed by themselves. Those arguments are then judged by experts, who determine their correctness. Authority and the means of implementing it are external to the students." 
 I am very excited about discovering Bryan's blog, because his thinking and most admirably, his doing, is very much along the lines of what I myself noted as the ideal in discussing the dangers of the classroom:
The learning process should leave the learner with an innate, ever-improving 'filter' of quality, to be able to evaluate for oneself, when encountering some piece of information, whether it is reliable, new, significant, important, or useful... [however] what is taught is often only what is already well understood, processed and neatly packaged. This makes it harder for students to think outside the box or understand the current limits of our knowledge, let alone how they can be extended...  The loss of autonomy and the presentation of processed information deprives students of the confidence to develop their own filter...

Bryan Meyer also has a nice set of recommended readings I plan to go through. I know of a few of them, but am unfamiliar with most.