"Critics argue that Khan’s videos and software encourage uncreative, repetitive drilling—and leave kids staring at screens instead of interacting with real live teachers."
I don't know where these critics have been, but that description sums up most of my math instruction in school.
I posted a comment there but it never showed up, I guess because it had some links in it.
I mainly recommended reading some educational research that could help improve the khan academy's work.
Especially research in math education.
See for example a report by Harold Wenglinsky called 'Does It Compute?" that found much better learning gains when kids did exploratory math activities and simulations and spreadsheets and so forth. When they did drill activities, their performance actually decreased.
http://www.ets.org/Media/Research/pdf/PICTECHNOLOG.pdf
Then there are these two books (which are free online), the first of which I'd recommend everyone interested in creating educational activities read:
In math, it helps to use gestures and embodied analogies and representations of many concepts. In other areas like science and history that helps, too, but also students have a lot of strongly persistent misconceptions that have to be brought out into the open in your videos or text, plus it helps to have supplementary constructivist learning activities like simulations, games, and modeling/design tools (including programming).
The thing that revolutionized my learning was wanting to create computer games. Not only does that motivate me to learn math, but it prompts me to start developing my own theories of reality to test within games.
This article got me thinking about the constructivist theory of education... and I realized that as a self-taught programmer, this theory hasn't worked for me except in very limited ways. I think constructivism works within a certain radius of mastered knowledge. No amount of 'fumbling around' is going to allow me to write a parser, or an operating system. Even for something that I have (or had) the motivation to do, which is to write a very simple learning game [0], the lack of knowledge was a show-stopper.
I think constructivism works for tasks that are just outside of your scope of knowledge. So if I really wanted to create a rails app, I could do that, since that's what I've been doing.
and indeed my mother must have learned that way in the rural one-room school for grades 1 through 8 that she attended during the Great Depression. It's good that now pervasive technology is helping teachers relearn that children who occupy the same classroom don't all have to literally be "on the same page" at the same time. That allows more able (or more motivated) learners to move ahead of the meager expectations of the United States curriculum,
"Critics argue that Khan’s videos and software encourage uncreative, repetitive drilling."
I agree that the Khan Academy exercises currently are just that, exercises, and have suggested here on HN that the Khan Academy exercise writers do the hard work of building genuine online problem sets.
Fixing the nature of the online exercises and turning them more into problems would help with one defect noted in the article: "Of course, kids who’ve grown up on computers are quick to spot the weaknesses in Khan’s system. They discovered ways to cheat on the drills: In the logarithms unit, for example, they noticed that the third multiple-choice answer was always the right one." But as the article points out throughout, what online reporting and pratice has allowed is for children who are already attending school anyhow to use their class time to discuss what are actual PROBLEMS for each learner, with interaction with a live teacher, saving drill for times of the day when the teacher is not available. That's not a bad improvement at all.
I should mention that in my own family we spend money on the ALEKS program
for our homeschooled children, and we find that very helpful. It has many of the same strengths of the Khan Academy program (but is not free, nor does it have audio explanations) with a more complete syllabus and more challenging problems. The "knowledge spaces" theory underlying ALEKS
is quite interesting and will probably have to be reflected by any complete K-12 mathematics program anywhere in the world. None of our children actively enjoy ALEKS, but they acknowledge that it is helpful for learning math, and I teach about the more cool aspects of math in the in-person math classes I teach through my local nonprofit organization.
Cognitive psychologists (unlike education PhDs, who rarely have a clue) argue that drills create a solid cognitive foundation for students to apply the material to other areas. "Deep" problem solving is a good IQ test, but doesn't improve IQ. "Shallow" problem solving lets students master the elements of the subject, which does increase their intelligence.
> "Deep" problem solving is a good IQ test, but doesn't improve IQ. "Shallow" problem solving lets students master the elements of the subject, which does increase their intelligence.
As you have defined the terms here, what you say makes no sense to me. It seems to imply as a direct corollary that solving deep problems does not make you any better at deep problems, whereas solving shallow problems does make you better at deep problems. I can believe the last half so long as it is only a supplementary form of training (a crude analogy might be of a virtuoso musician practicing scales), but the first statement beggars belief and is contrary to all my experience.
Professional mathematicians peak in their 20s, just after they stop solving "shallow" questions, and start poring over "deep" ones. Pianists peak just before their fingers are unable to physically hit the keys at the rate they used to. But then, unlike pianists, professional mathematicians have something of a conceit that playing the same scale a second time is a trivial waste of time.
Of course, intelligent people are drawn to "deep" problems. Being willing and able to solve "deep" problems is practically the definition of academic intelligence. Actually, most people like "deep" problems more than shallow ones, but only intelligent people have the mastery of the basic techniques required to takle the deeper problems.
I'll admit this is all very much [citation needed] stuff, though. I'm not certain.
Professional mathematicians peak in their 20s, just after they stop solving "shallow" questions, and start poring over "deep" ones.
Citation? Wiles proved FLT at age 40. Perelman proved Poincare at age 39. Richard Hamilton was 50 at the time.
But even with that great mathematicians START doing deep problems early in life. By the time their 25 they've been doing deep problems for a decade at least. Terrence Tao, for example, was doing calculus at age 7. He got a gold at IMO at age 13. These great mathematicians like Tao, Goedel, Neuman have long gone beyond grill and drill math before age 20. So the notion that they just recently stopped solving shallow questions is absurd.
> Professional mathematicians peak in their 20s, just after they stop solving "shallow" questions, and start poring over "deep" ones.
This is a ridiculous myth. As a truism, person's creative intellectual output is greatest when they are working hardest and have the greatest number of tools of creation available to them. I could go down a long list of mathematicians (famous and not famous) whose most influential work was done later in their careers.
There is support for this in skills like Martial arts, Music and many art forms. Repetition has been shown to create abstractions inside the brain that eventually help understand the more "deeper" problems better -- though there is no significant evidence. The fact that many learning forms (mostly eastern) have used it for such a long time suggests that we cannot completely dismiss it outright.
Could you tell me which cognitive psychologists say drills create a solid foundation?
I agree that cognitive scientists say that extensive background knowledge is necessary for a solid foundation, however having background knowledge does not necessitate drills. I haven't seen any studies that specifically say that pure drilling is good, and instead have seen the complete opposite. All I have read also enforce my beliefs that mindless drilling necessary isn't good.
tl;dr: The drill work is fine, as long as its motivated. Though it's obviously much more nuanced than that.
"An extension of this argument is that excessive practice will also drive out understanding. This criticism of practice (called "drill and kill," as if this phrase constituted empirical evaluation) is prominent in constructivist writings. Nothing flies more in the face of the last 20 years of research than the assertion that practice is bad. All evidence, from the laboratory and from extensive case studies of professionals, indicates that real competence only comes with extensive practice (e.g., Hayes, 1985; Ericsson, Krampe, Tesche-Romer, 1993). In denying the critical role of practice one is denying children the very thing they need to achieve real competence. The instructional task is not to "kill" motivation by demanding drill, but to find tasks that provide practice while at the same time sustaining interest. Substantial evidence shows that there are a number of ways to do this; "learning-from-examples," a method we will discuss presently, is one such procedure that has been extensively and successfully tested in school situations."
...
"Real competence only comes with extensive practice. The instructional task is not to "kill" motivation by demanding drill, but to find tasks that provide practice while at the same time sustaining interest. There are a number of ways to do this, for instance, by "learning-from-examples.""
Ok we're on the same page then. I fear that the word "drill" is often linked to mindless repetition without caring for motivation. However, as that paper says ("The instructional task is not to "kill" motivation by demanding drill...") it emphasizes the importance of motivation in the process, which is great. I well understand the role of practice in competence development as well.
> That allows more able (or more motivated) learners to move ahead of the meager expectations of the United States curriculum, [...], an improvement that is long overdue.
I suspect that a greater focus on advancing rapidly through the curriculum would be a mistake, no matter how well-intentioned.
If we're talking about mathematics applied to everyday life, what's missing are the fundamentals that you're supposed to complete by fifth or sixth grade but most people never really master.
If we're talking about preparation for a mathematically intensive course of higher education then I also don't see the rush. There's often a great disparity between the amount of mathematics "known" among the smarter math freshmen entering university. Some will have read ahead to the beginning graduate level and others will have more closely followed the curriculum's set pace. But given a pair of equally smart students of these two types, they will generally have achieved some level of parity in the extent of their knowledge around the middle of graduate school, their areas of specialization being equal.
The explanation for this phenomenon is surely complex. But I propose that there are two main reasons. The first is that when you race ahead you're likely to leave some gaping holes in your understanding--filling those holes means long, hard work as compared to the glamorous alternative of racing ahead to behold new and exciting mathematical vistas. The second reason is that you've become much better and faster at learning new areas of your subject by the time you get to that level, so it can be time efficient to put off learning until you're more truly ready for it.
What you say is both right and wrong. It is wrong because sometimes children already know the material and would like to move on to other things. On the other hand, letting people race ahead sets up certain unhealthy incentives for people not to properly master their subjects.
Yes, that is what I was trying to convey. The final part of mastering a subject often consists of hard, unglamorous work. Addressing your own weaknesses is never much fun, even if you know deep down that the pay-off will ultimately be worth it.
Providing problems automatically on a website like Khan Academy is technically very challenging, because problems are hard to write. Currently, the many exercises are generated automatically with random numbers every time but such a scheme is impossible with harder, deeper problems – instead the problems need to be written thoughtfully and carefully, something that is hard to do on a large scale. In addition, answers to problems are usually significantly harder to verify than answers to exercises and so would be hard to implement on a large scale like Khan Academy is trying to do.
Like you point out in this post, problems are probably best left (for now) to individual teachers who can spend more time and effort on individual students than is currently possible on a larger scale.
Creating problems may need to be done carefully, but there is no reason it can't scale. Khan and a few others can create thousands of good problems and distribute them to every internet user.
Individual teachers creating problems adds little value. Timmy in NY can do 1000 of Khan's problems and so can Samir in Pune. What does it matter if Timmy's problems are different than Samir's?
I don't know where these critics have been, but that description sums up most of my math instruction in school.