The headline reminds me of Feynman's account of teaching physics in Rio. Every student could stand and deliver a verbatim retelling of all the definitions in the textbook. They had just covered polarized light. The classroom had a beautiful view of the bay, with sunlight glinting off the water.
Feynman asks for and gets a recitation about polarized light, which includes a bit about reflected light being polarized. He then turns towards the window and asks the class for an example of polarized light. The whole class is flummoxed. Everyone could recite, but none of them could apply.
EDIT: I remember the day, being a TA for a Comp Sci 101 class, when I realized that a lot of the most vocal and active members of the class were more interested in the mechanics of lectures and tests than the material. It was all a game. I put stuff on the blackboard and in the handouts, and they spit it out on the test sheet. We have become a society of appearances and bureaucratic mechanism. Most college students go through the motions and get their paper, then go home thinking science is just more made up bullshit like politics. These folks laugh when you talk about evolution as reality, and can only reason abstractly at the level of a grade schooler. Many of them graduate, vote, and get promoted to positions of authority.
I'll never forget when I was interviewing a candidate for a junior C# programming position and the conversation went as follows:
me: Can you tell me what a delegate is?
candidate: <text-book perfect definition, absolutely flawless>
me: Great! Can you tell me when you would ever want to use one?
candidate: <silence for about 15 seconds> I don't know.
It was so bizarre and I still don't fully understand how you can understand the absolutely perfect theoretical definition with having no, literally not one, idea what you're learning.
Personally, I simply cannot _recall_ anything unless I understand it at least superficially. I may be able to _remember_ if prompted, but my memory is simply organized around my own definitions and understanding.
Maybe us engineering types are hardwired this way. I suspect, however, that non-engineering types aren't wired this way due to poor parenting, weak schools, and fundamentally broken reward and punishment systems.
I simply cannot _recall_ anything unless I understand it at least superficially
Me too. That's why the wheels came off my math skills when I got to differential equations. Everything up to that point I'd been able to form a picture of in my head. For example, calculus is just Newtonian mechanics, so that's got a clear analog in the real world.
But Diff EQ was just too abstract. I was eventually able to pass it with the help of a couple friends pumping me full of formulas and procedures, but I never really understood, and so I never could really apply it on my own.
I took a linear algebra class and they explained nothing. Sure they'll prepare you for their tests, but otherwise their material was way too abstract. Not until I watched a lecture by Gilbert Strang on iTunes U did I finally understand what an eigenvalue/vector is and what the hell the matrices represent.
So mcherm, I'm pretty sure you don't need any enlightenment on this, but I'll try to peer teach this to whom it may help.
So let's say we have a vector which is represented by the matrix A of any dimension (any # of columns, each column a vector). If we multiply this matrix A with an arbitrary unit vector (considering just the direction in the given dimensions) B, then we'll have a certain set of possible resulting vectors (A X B). The eigenvectors are whenever the same unit vector B lines up in direction with (A X B)--where both vectors are dependent on B, and the eigenvalues are the corresponding scalars which must multiply the unit eigenvector to match the resulting A X B vector. The eigenvector does not have to be a unit vector, but since any multiple of the eigenvector can still be a valid eigenvector, you probably just want the unit vector or at least the lowest whole number reduction. If you reduce the eigenvector, the eigenvalue must be changed too.
In equation form: Ax=Ωx
Where A is a matrix you are trying to transform, and x is the possible eigenvector, and Ω is the eigenvalue.
And the matrices are simply a different form of a set of equations while still preserving their relative properties. The matrix form allows you to easily manipulate the original set of data with matrix operations. I guess the matrix can also just represent a set of column vectors (which are just a direction in a space with a magnitude) too.
I was in that boat for a while being both of a similar mind and in a similar situation. I got over it by the simple work of being absolutely pulled through the ringer by a diabolical prof on Abstract Vector Spaces.
(btw: I love that professor today. He was just underhanded enough to do what we all needed.)
I'm utterly convinced that pure math needs to be taught at a high level for anyone in advanced science/engineering. It hurts for a while, and then you look back and everything is clear, simple, and bright shining.
I've also heard this just keeps happening if you keep on with math.
I have had the same experience in a lot (all?) of my math classes. I never really understood integrals and differentiation until I saw them come up in a physics class. Once I saw the practical application of those things, I didn't have much of an issue in actually solving them. That's the primary reason why I hate math courses. I know there are practical applications to everything I'm learning, but I just can't see them.
When using math for real world applications, you essentially have three steps. First translate the problem into a math problem. Then manipulate the math problem as needed, then finally convert back the real world problem. But in most classes, the second step is the only part that is covered. This is the part where most people have problems, since people are wired to deal with the world around them, not some abstract math world. Trying to ground the math concepts to real world situation is difficult and complicated, which is probably why this step is skipped.
Very true, they are incredibly useful. But as taught in most class rooms the application is far removed from the instruction. A differential equations class is often not much more than showing some techniques and then assigning numerous quantative problems that use those techniques.
So, taking a differential equations class by itself can make it seem abstract and with limited connection to the physical world.
I simply cannot _recall_ anything unless I understand it
I'm chiming in as well because that is exactly how my mind works. I can remember very little trivia or material I am fleetingly familiar with. However, I do extremely well when applying concepts, and later I can remember intricacies extremely well.
I also have great difficulty presenting material that I don't fully understand, yet when I understand it I am a great public speaker.
I have found that preparing to explain something makes me think and research it more, and understand it better. Some things that I sorta-knew how to work and where they come from became my own intellectual possessions, instruments, and later part of the toolbox, once I got to know them good enough to pass the workings on to someone else.
That third party needn't even exist, once you get the hang of learning new and lesser-understood stuff that way.
My learning style is exactly the same way, I can barely memorize four-character strings unless I can find some sort of mnemonic pattern, but abstract concepts are quick and easy. In a surprise twist of fate, that's why engineering was utterly impenetrable to me in school, while mathematics (my major) was a breeze. I think it has to do with the method of teaching. My required intro engineering courses were rather practical, so I couldn't find enough theory to wrap my head around.
There's evidence that you're right about the correlation with upbringing. Someone did a study on elementary school children, teaching them basic one-digit multiplication, and the concepts of place value, and then had them invent their own algorithms for long multiplication rather than rote-memorize the standard one. The methods they invented were invariably less compact and required more work than the standard method, but the students also performed well above average on standardized tests for multiplication.
I suspect that the high-performing individuals in the "softer" disciplines like literary analysis also function in the same way, just on fluffier concepts and less rigorous abstractions.
I remember the day when I figured out that multiplication is iterated addition. No one had ever bothered to explain that. We were just told to memorize tables. I was at the bus stop. "Three times three. OH! Add three together three times! Don't you get it? That's why it's called TIMES!" My fellow grade schoolers just looked at me. (I grew up in a truly podunk town!)
What? Anyone who understands the evidence for evolution should accept the theory, because it's just that compelling. If there is another explanation for all of the data & experimental confirmations then let us know, I'm sure you've got a paper in Nature awaiting.
Some people who are otherwise reasonably bright have real trouble understanding systems. Interestingly they tend to accept claims about systems that they don't really understand that fit with their social/political associates. For example, liberals tend to accept evolution but question free-market economics because that is what their associates do, and vice versa for conservatives. Those that can understand both tend to become fairly libertarian, though remaining liberal or conservative on many specific issues.
I'm the same way. As a student, a trick I used was to explain the material to an imaginary person in plain English(+). The process of formulating the explanation in your head does a lot to cement your understanding of a subject.
(+) In my head, not, say, out loud while waiting for the bus.
For me I am the exact opposite. You might ask me what a delegate is and I might say, "Uh... I don't remember." Yet it is something that I use every day in my programming, I just don't usually think of it as being a delegate.
I was fascinated when I took my first C++ class and learned that using a pointer was called referencing and dereferencing. I had been using C++ for years on my own and had always thought of it as:
x = &y; //X equals the address of Y.
x = *y; //X equals the value at the address Y.
To me that made more sense than calling it "referencing" and "dereferencing".
Which seemed much more logical, but led to trouble when the internet came along and I started having thoroughly confusing conversations with other people.
Exactly the same. I also never really thought that pointers were difficult yet it seems to be a major stumbling block for many beginning C programmers. Perhaps having already done assembly language programming before I learned C was the reason.
Right, in that same class most of the rest of the students simply could not seem to understand pointers. They asked me, a fellow student, questions like "Why would you ever use pointers?" The thing was, I think my lack of "jargon" made me explain the answer better than the professor did. Hence the other students asked me, not the professor.
It seems to me that fancy terminology such as "referencing" and "dereferencing" is not only not common sense, but it also makes it harder to understand things when you are starting out. Perhaps as professional programmers it makes it easier to talk about algorithms between each other, but jargon is generally for the sake of exclusivity, not for the sake of convenience. I would even go so far as to suggest that anything that can be said with jargon can probably be said better with simple language.
What beginning programmers are learning these days is jargon, and the definition of jargon terms, but they aren't learning what the meaning of those terms is.
There is a big difference between learning the definition of something and learning its meaning.
As you say using it early into an intro programming class may be a mistake, but it's soon useful in that you can say 'don't dereference b, it might be null' instead of something like 'don't attempt to read the data located at address b, b might be a null pointer'.
Are you sure about 'referencing' though? I've never heard it in this context. It could be confusing given how C++ has references has well.
Sure, that's like people being able to talk long before they learn what nouns and verbs are. A matter of learning to use something versus learning and remember terminology to describe that something.
Yeah, I'm not a big fan of the term defererencing. There ought to be a better word for following a reference.
(This question is of practical interect to me -- I'm currently designing a language that has a dereferencing operator deref() and am wondering what better name to give it.)
Sometimes it works the other way round. One professor solved an equation in a class by integrating with respect to pi. The students could not believe their eyes: "You can't do that!", they said; "pi is a constant!". The professor explained that once you have expressed your problem in a formal system like mathematics, you can do anything that is allowed in the rules of that system, even if the intermediate stages make no sense with respect to the original problem, and still obtain a correct answer.
[This story is from E T Jaynes 'Probability theory, the logic of science'. But I don't have it to hand, so I may have the details wrong (and I can't remember who the prof was - it wasn't Jaynes).]
These folks laugh when you talk about evolution as reality, and can only reason abstractly at the level of a grade schooler.
Since you tossed this subject in, how about folks that disagree with (macro)evolution but have a Ph.D. from MIT and currently work as a Nuclear Research Manager for NASA?
I've met this guy and he does not reason like a grade schooler. Also, he noted that many of his colleagues agree with his position regarding origin science.
how about folks that disagree with (macro)evolution but have a Ph.D. from MIT and currently work as a Nuclear Research Manager for NASA?
I followed the link you kindly provided. I see this man's higher education was not majoring in subjects most likely to expose him to the evidence for biological evolution. Nor does the link list any peer-reviewed papers of his on any subject related to the evidence for biological evolution. Macroevolution is a fact
in the same sense that nuclear fission is a fact. Neither is routinely directly observed with understanding by laymen, but both sets of facts can be demonstrated by experiment and are consistent with broader bodies of verified scientific knowledge.
I brought him up as an example that there are certainly people out there that don't agree with macro evolution and can reason well beyond that of a "grade schooler." I was not trying to allege that he himself has done any research directly refuting origin evolution. Perhaps if origin evolution could come up with scientific theories that are falsifiable, we'd see more refuting research.
I brought him up as an example that there are certainly people out there that don't agree with macro evolution and can reason well beyond that of a "grade schooler."
Fair enough. And I brought up the evidence for macroevolution,
in relation to the main point of the thread-opening submission, to show that college-educated persons with normal curiosity, reading ability, and access to the Internet should be well convinced that macroevolution is a fact, in the same way that nuclear fission is a fact. Perhaps the problem of someone who doubts macroevolution (as, in fact, I was once taught to do) is not a problem of reasoning ability at all, in agreement with you. (I thought my reasoning ability was just fine, and some forms of school testing including the GRE suggested that it was, before I became convinced by the evidence for macroevolution. But I had a crucial lack of curiosity when I wasn't looking up the evidence
to use my reasoning ability to evaluate that evidence.)
Now I'm curious about the puzzling pattern of upvotes and downvotes here. I think the concern about science education expressed in the thread-opening submission would be a general concern of HN participants, and I would think that a comment that posts a link to a source of reliable scientific conclusions in such a thread would not be considered a detriment to the quality of discussion on HN.
Perhaps if origin evolution could come up with scientific theories that are falsifiable, we'd see more refuting research.
Demonstrating my curiosity in what I hope is a friendly way, what kind of theories do you have in mind? What would be a way to falsify a theory (from any point of view) about origin of species?
> Most college students go through the motions and get their paper, then go home thinking science is just more made up bullshit like politics.
But I don't see how anyone can come to a conclusion that "science is just more made up bullshit" when the application of science can be, and has been, practically demonstrated to be accurate, in cases such as sending people to the moon, a lander to mars, and even detonating an atomic bomb.
Sending people to the moon and the atomic bomb were as much political as scientific. Without politics, I doubt either would have happened when they did.
The topic of politics has no bearing whatsoever on my statement of incredulity that any sane person could draw such a conclusion as to think science is bullshit. The text could have read "...thinking science is just more made up bullshit like [Santa Claus]" for all I care.
I once had a girlfriend who thought that way. Thought that going to the moon was a complete waste of resources because it "made no real difference in anyone's life"
I spent most of my college years doinking around with hacker wargames and trying to hack kernel code, albeit not well. I didn't get graded on that but rather on how well I could put together a resume or whatever.
I realized at that time that my "grades" would suffer if I didn't "work" harder at it but I made a conscience decision to hack more. I did as much as was required to graduate and did what I wanted the rest of the time.
This is really excellent ... and it makes you think: we (US) used to have one-room schoolhouses where the older kids helped the younger kids. Another thought: a lot of the differences between the prof and students would be removed by physical discovery rather than lecturing.
I never liked lectures. What kept me going was talking to people and DOING stuff. And those things I retained for DECADES ... the rest was gone in a year or two.
For anyone who this video resonates with, I suggest watching the highly entertaining Bollywood hit 3 Idiots, about three mechanical engineering students a a prestigious indian engineering college, struggling against their autocratic principal :-)
I think it's slightly unethical to take it word-for-word without giving a hat tip.
That said, this is a rather remarkable video if, like me, you don't know much about the theory of education (I showed it to a smart friend who teaches undergraduate literature and he said that it was all old hat to him.)
What really stood out to me was the hard data behind Mazur's conclusions -- it's "common sense" that students do better when engaging in the process of learning, but I had never before seen it so convincingly demonstrated.
Sorry! I waited a day for you or JabavuAdams to post the link, but since you both didn't do it, I posted it myself, because I felt this was good enough to be posted here !
When I was in physics grad school we used the Force Concept Inventory for assessing the intro level physics classes. It was always very interesting and showed that they were able to greatly increase understanding over the course of a few years by focusing on a few areas.
I feel that (introductory) physics education has benefited greatly from education research, partly because data can be easily collected and at universities you can adapt and iterate on a semester (or annual) time scale.
It would be nice if this model could be adapted to other areas, such as graduate physics, but the normal response is that grad students are supposed to be responsible for their own learning...
That's true, but it puts teachers at a post-primary-school level in a real bind. It's hard or impossible to take a subject like physics or calculus (or even algebra, or English literature) and teach it from first principles in a semester or two, especially when most of the students are not very self-motivated or interested a priori in the work.
So what do you do when you have a class full of students who have already been basically failed by the system and who understand none of the foundations of your subject? If you fail them all, you're the one that is likely to get the ax, even if it's true that none of them understand your material. Most teachers resign themselves to doing their best and administering fairly traditional tests and classwork, so that hard-working students can pass even with no understanding.
You're right, which is why teachers/schools/education authorities shouldn't be setting the exams. The purpose of the exams is (or should be) as much to check that they're doing their job as it is to examine the children.
True, but in practice this isn't what happens. What usually happens is that low performing teachers (or school systems) lobby to have the standardized exam watered down.
This is a fundamental problem of agency costs. The only people who gain from having a good testing system are the high performing students, and we obviously can't put them in charge.
Teach more slowly. If I could, as a student, take courses that were "as long as necessary" rather than limited to a semester schedule, I believe I would be a lot more satisfied with my education thus-far.
I don't see any point in graduation, really. Why stop learning? The ideal school curriculum, in my mind, would have four unassisted co-op terms (i.e. two years of real, normal work) followed by one or two terms of instruction, repeated in a cycle, for as long as you may live. At no point would you be handed a certificate and kicked out of the building; you'd just keep coming back and learning more stuff (presumably in a myriad of unrelated subjects), then going back out and making use of it.
Sometimes you really understand something only in retrospective -- and after you learn a lot more.
Your understanding of basic science improves significantly when you see 'how things fit' in a much larger framework.
Moreover, as your studies progress (sometimes much after the exam) you see new applications of what you have learned. And this sheds new light on the subject and you get deeper understanding of the subject.
But, really, taking what he's doing to the limit (he even says so himself) is just re-creating the Socratic method. (Though choosing the questions is hard work, as he notes.)
And he can't do that, given his large class sizes (one of my classes at Harvard was 700+ people, Ec 101 with Samuelson from MIT), so he uses this peer-teaching approach to work around the limitation that he can't sit down with a small group of them and work through the questions.
That's why we are sending some of our home-educated kids (those that show the interest in and capacity for a rather intense education) to Thomas Aquinas College in Ojai, CA.
It's probably the only school in the world that uses 100% Socratic method for all courses (each no larger than 17 people), with no electives all 4 years, outside of St. John's of Annapolis/Santa Fe (on which TAC is modeled to some degree).
(Yes, it's an unabashedly Catholic school, but that doesn't diminish the intellectual rigor in the least. It's also in one of the most beautiful spots in the world, in the foothills of the mountains north of LA, near Ojai, which is where the 50's film "Shangri-La" was made and which is still a spa/resort area today.)
arguably has a more discussion-oriented approach to learning, definitely has smaller classes, and has a rich supply of elective classes. And it's for high school-age students rather than for college-age students. We are also a homeschooling family, but we thought long and hard about sending our oldest son to Exeter (which also has great financial aid) before continuing to homeschool him. We may revisit the issue of how to do secondary education with each of our other three children. Our oldest is now applying for colleges--every college on his application list is a research university with strong programs in mathematics and computer science (and, yes, lots of elective course possibilities).
A little exposure to the "Lord of the Flies" might be good seasoning, provided he's been given a few psychic defense lessons ahead of the time. (I was thrown into the boarding school situation with no defenses at all and had to cook up my own. A little guidance would've been worth a lot.)
My experience with a bunch of Exeter kids in my college class (admittedly a long time ago--'72 to '76) was that they were burned out.
They had just completed what was the equivalent of most college degrees today, and were looking at another 4 years of even more intense work. Most of them just sat around their oriental-draped rooms in deep club chairs and smoked weed. ;-)
There's nothing invariant about homeschooling as practiced by all families who practice it that guarantees not learning about social interaction. Homeschooling can be a very good environment for socialization.
I'm more familiar with St. John's College, as I dreamed of attending had not the cost been so prohibitively high.
http://www.sjca.edu/
But the one-track curriculum is based off the trivium and quadrivium of medieval education, where you don't have electives but regardless, you get a well-rounded education because they're preparing you to think and inquire. So every student studies Greek, Latin, French, Western classical music, mathematics from the source (like Euclid's Elements and Newton's Principia), physics from the source, chemistry, history, etc. No modern textbooks rehashing information, so theoretically, no passive learning. So this curriculum isn't really a straitjacket.
The one downside is that you're not getting a very multicultural or contemporary view, given that you're reading the Western canon largely pre-World War II. But it's not such a big problem if you again realize that the idea with this education is that you're being trained not in subjects but in the ability to go out and find what you don't know in the classroom on your own and to discuss these new ideas with other people.
I'm actually quite a fan of the "one curriculum" approach (and schools which implement it seem to produce more successful graduates, on average). One of these days I even plan to write up why (and then get piled on for suggesting that there were some good things about the "medieval" university system).
I think there's some significant innovation in scaling the Socratic method here. I do not think it is clear a priori that such a thing is even possible.
Law schools with very large class sizes indeed claim to practice "socratic" instruction, which I found quite enjoyable as a student. Classicists think it's an abuse of language to call what law schools do socratic dialog, but it is definitely much more interactive, even in the largest first-year classes, than what is usually done in undergraduate courses in almost any subject other than modern languages.
For people interested in this stuff, Howard Gardner's book _The Unschooled Mind_ collects similar results from every discipline, from computer science to history.
Does anyone know what textbook he's referring to as the one he used as a base for his original lectures? Sounds like 'Weidner and Selz', supposedly out of print?
http://www.youtube.com/watch?v=WwslBPj8GgI#t=22m53s
Feynman asks for and gets a recitation about polarized light, which includes a bit about reflected light being polarized. He then turns towards the window and asks the class for an example of polarized light. The whole class is flummoxed. Everyone could recite, but none of them could apply.
EDIT: I remember the day, being a TA for a Comp Sci 101 class, when I realized that a lot of the most vocal and active members of the class were more interested in the mechanics of lectures and tests than the material. It was all a game. I put stuff on the blackboard and in the handouts, and they spit it out on the test sheet. We have become a society of appearances and bureaucratic mechanism. Most college students go through the motions and get their paper, then go home thinking science is just more made up bullshit like politics. These folks laugh when you talk about evolution as reality, and can only reason abstractly at the level of a grade schooler. Many of them graduate, vote, and get promoted to positions of authority.