However, I have had some seriously worrying conversations with people who have some deep, deep misconceptions, and who believe they have been vindicated by some parts of some of the videos. Again, correlation versus causation, etc., but some of the things these people believe they've learned from the KA, and believe must be right because they (think) they've seen it on KA really worry me.
I agree with Sal when he says:
We believe that we are in the early days of
what we are and feedback will only make that
better. I agree with you that no organization
should be upheld as a magic bullet for education
It's a brilliant body of work, certainly ground-breaking, and potentially revolutionary. But it has its problems, and denying that doesn't help.
What kind of things?
Are you suggesting that flesh-and-blood teachers or brick-and-mortar schools are less likely to spread misinformation or leave students with misconceptions? I can tell you a hundred false things I was taught by "real" school, and some of them are probably just things I think I was taught.
Goals aren't results. Compare real to real. The real competition isn't the mythical fantasyland where everybody learns everything perfectly and retains it forever as a result of the gentle teachings of the schoolmarm. The real competition is the system that produces the real people you really run into as you go about your day, doing real things. If online education in its currently very early state even manages to sometimes match the real world, at one-tenth the price or less, you can imagine what another ten or twenty years of refinement is going to produce for the online model.
And in that real comparison, online will definitely not always win today, and will always have weaknesses. Shop class seems unlikely to work well online. (But then, the real school systems shut down those classes 20 years ago....)
Truly the greatest enemy the $REFORM_MOVEMENT faces is not the $EXISTING system, but the way that people secretly substitute the real system for this mysterious magical perfect system based on the assumption that the stated goals of the current system are actually its result, and compare online against the myth instead of the reality.
Possibly, because when you leave school you can realize your misconceptions and learn from other people, as well as pointing out other people's misconceptions. But if everyone uses Khan Academy, everyone will end up with the same misconceptions.
Also, online learning wins in the limiting case where courses have been iterated and feedback / FAQs taken into account. In a brick and mortar institution, "institutional memory" is effectively lost as professors retire or shift interests (if it ever existed in the first place) and their replacements introduce their own slew of idiosyncrasies, poor methodology, and misconceptions. In contrast, an online lecture never suffers this degradation. Sure, new lectures may replace old lectures (presumably at different sites), but both will be available for a time and during that time, students will pick up on inconsistencies. That's not true of brick-and-mortar lectures, where the only source by which a student can verify the material they are presented is the textbook or its alternatives. Combined with the tendency of courses to discourage reading the textbook (it's almost always a less point-efficient use of time to read textbook chapters than to do homework, practice problems, and go to office hours, which provide information better matched to the material that will be tested on), the status quo is a recipe for failure. My experiences confirm that this recipe for failure produces failure (in the pedagogical sense).
The third factor that favors online learning with regard to misconceptions are its twin killer features: the pause button and the rewind button. These reduce self-inflicted misconceptions tenfold.
So, sp322, I think the opposite of what you stated is true. I would welcome a conflicting opinion, because usually when I go on rants similar to the above three paragraphs all I get are nodding heads.
I guess some people are suggesting that Khan Academy is the ultimate in education and that everybody should use only that and nothing else?
The top-poster's point was about a very specific type of problem he's seen with some people watching some of Khan's videos. He does not even hint about the prevalence of the same problem in traditional schools.
He found that just showing a video of the laws of motion made people more confident of their answers on a physics quiz, but just as wrong. They would actually have false memories of the video confirming their prior knowledge. What worked was first debunking misconceptions and then replacing them with the truth, even though those videos were rated much more confusing.
We're not teaching blank slates, and if you teach as if you are, you end up with what you (and Derek) described.
You need to to drag someone through the mental struggle of unseating their beliefs.
My favorite technique is to find the source of the misconception, and highlight it. Instead of attempting to dispose of an entire complex thought, pick it a part, find the mistake ("you are confusing 'velocity' and 'force' -- they are different things."), and rebuild the correct explanation using the patched piece.
With that said, I think that KA is an excellent addition to the pieces of the educational puzzle. I've used many of his videos to supplement teaching and textbooks, both good and bad.
At the same time Khan Academy's video lectures cannot take into account the prior knowledge of students but for what other videos have been downloaded at Khan's. It cannot take into account students' background, wishes, problems, disadvantages, and so on. There is no interaction, there is no conversation going on between teacher and pupil, pupil and pupil, and in the social environment of a class.
Furthermore, most videos and exercises seem to focus on repetition, rote learning, and giving correct answers. There is no room for exploring alternatives, errors, hypothetical situations, and what not.
Over-all, and most importantly, I wonder if Khan Academy does support/instigate/promote deep learning of the topics covered. And is that actually possible the way Khan Academy is set up?
Observe that Khan himself doesn't claim nor suggest so, but many who hail Khan as the savior of education seem to equate education with learning and, as a result, claim Khan's revolution in education as a revolution in learning. To me, these aren't the same and this pollutes the discussion between educators and fans.
To summarize: I think Khan's revolutionizing education but he isn't revolutionizing learning
Khan Academy is interesting to me not really because it's a revolution in learning, but because it's another modality that is better for some people. Some people learn well from written material, and those people have traditionally been the successful autodidacts, working their way through textbooks and tutorials. But perhaps different kinds of materials can cater to other people, with people working their way through a lecture series instead of a textbook. That'll probably have many of the same pros and cons.
Where i was taught we still had the exact same decimal problem khan has...
The best way to deal with it is through continuous refinement until criticisms are few
> The best way to deal with it though is
> continuous refinement until criticisms
> are few
I am, almost without exception, deeply disappointed by the knee-jerk reactionary defensiveness you generally find in these things. Conversely, you go visit Wikipedia itself and it has insightful comments about how its user base tilts the content away from the global south, women, the poor etc. and constructive ideas for how to measure and ameliorate such issues.
I think it's only natural for people to react to such uninformed, accusatory criticism in the way they do. It reminds me of John C Dvoraks anti-Mac columns from years ago, which he later admitted were professional trolling.
The question shouldn't be "How well does Khan Academy teach" (well, maybe someday). The important question right now is "Is Khan Academy a better learning tool than most high school and college teachers?". In my experience, the answer overwhelmingly is "yes".
I did have some truly fantastic teachers/professors on the way to a STEM degree, but they were rare. Most were terrible to mediocre, even at a semi-prestigious university. Khan Academy has greatly reinforced my understanding of some subjects, and it's been exceedingly easy to learn some new subjects.
So yes, it doesn't stack up to the top tier teachers and probably never will. But that doesn't matter, because there aren't all that many fantastic teachers and Khan Academy is better than the rest.
I try not to ascribe unstated motives to people, but I really wonder at the smell of pedantry and elitism in an article like that.
Khan offers a free educational service that is very popular. That is a Good Thing(tm)... PERIOD.
People are logging in to his site to learn things. Khan has created a whole model for learning online that has some people enthusiastic about learning more than they could by sitting in class -- yet the criticism from this article is that the Khan videos aren't ideal in using the latest buzz-wordy techniques approved by academia?
Maybe the authors of the articles/papers could interact with Sal Khan and offer help. Or if the authors really felt strongly about their qualifications vs Khan's, maybe they could offer free online teaching videos that were more "correct".
This type of headline sensationalism designed to take down someone doing a good thing is just sad.
Yes, because when something is popular and free, its got to be good. cough religion anyone cough.
If that is considered to be the measure of quality these days, our society is doomed. Education is an important thing, ignoring it and treating this one man as some "messiah" of education when he clearly introduces wrong and obtrusive concepts for the sake of "getting it out to the masses" is not education.
It would be like if Khan Academy was giving out free meals to homeless people and McDonalds fabricated reports on how the food was unhealthy and used their influence to get the Washington Post to carry the smear and add to it.
wrong and obtrusive concepts
Now who is showing a religious level of bias?
I'm a pragmatist. If something is free that I need, I use it. If something is for-pay that I need and the price is justifiable, I use it. My commercial software architectures often mix proprietary and open source solutions and I don't make excuses for either business model.
One thing about free, you can always get a refund on your investment and you have to really suspect the motives and the character of someone who attacks "free". Not real surprising that the attacker in this case was a for-profit competitor, was it?
There is no religious bias, its a difference between a ineffective and wrong way vs. an effective right way. The professor performed no smear against Khan Academy, but the fact he only saw two videos that cause confusion in basic math concepts, it should make one skeptical, especially when someone like you justifies it on the basis of popularity, pragmatism and "minutia".
It seems like you read one article from the Washington Post that wasn't written very well and from that one article, deduce that WP is on some smear campaign. Absurdity of the highest order.
I've just finished a BSc Physics-Math at a UK "Top 20" University. The standard of many lectures was very poor with only a few exceptions.
Without Khan Videos I wouldn't have grasped many of the concepts. (The lecturers often presented the formalism without really going into the concepts - I often wondered did the really understand the material themselves)
Apart from online material the only other self-study stuff I found useful for the degree were course books from Open University (OU). Excellent stuff.
Many good instructors can't draw a line between KA (or any other disruptive solution) and the absolute best, gold standard, scientifically-proven best way to teach (usually their favorite methodology).
The fact is that he's better than most of the teachers out there. It's a rising tide: if your teacher has a Presidential Award for Excellence in Mathematics and Science Teaching you probably don't need Sal Khan.
Khan will put the video out there and see how people react to it. He perceives this to be a better approach than incorporating results of quality research projects into his instructional decisions. In the age of No Child Left Behind and its mandate for “scientifically based research” as the foundation for classroom instruction, this seems lazy.
Yes, Khan is doing his own scientific research, on the theory that he does a better job than the educational establishment. This might be arrogant (given the quality of educational research I've seen, I don't think it is), but it isn't lazy.
As for the claims that Khan does a worse job than other teachers, no data is provided to back this up. Just vague claims that if he did something different, he might get better results.
While PCK is a "buzzword", it does represent something very real in learning: That Instructors (be they amateur or professional) must understand the hidden challenges of the material they teach. Specifically, in ensuring that misconceptions do not propagate and embed themselves into a pupil's mind. In math, misconceptions can be particularly troublesome as the facets of mathematics, particularly at the elementary level, carefully layer on top of each other.
If you want an example of this that falls into the sphere of stuff closer to HNs, look to Zed Shaw's Learn Code the Hard Way series.
Shaw puts exercises and examples in front of readers designed explicitly to prevent the reader from developing weird misconceptions about the way a programming language (or programming in general) works. That's often why readers will say things finally "clicked" when they work through his material. Shaw may not be familiar with the buzzword term "PCK" but he definitely recognizes the challenges in teaching each concept he presents to the reader. You can see part of this come through with his critique of K&R C.
That's the very essence of scientific experiments. You don't always need a "hypothesis" to do science. (Or, if you like, the hypothesis is "This video will teach students to do X.")
While it's not controlled experiments, it's about as close as you can get in education.
In the debate about khan academy, there's an important fact that is continuously missed, it's not about effective instruction vs. ineffective instruction; it's about less effective vs. more effective.
The khan academy way may be better than trying to learn from the lower X% of math teachers out there. It's highly unlikely that it's going to be better than the top Y%.
Things like Khan academy will not "fix" education. Anything that standardizes and automates education will only do one thing: insure mediocrity. It will protect against the bad while eliminating the possibility of excellence.
In 2010, the US government spent $900B on education. If Khan can reduce the number of teachers needed by even a few percent, that's a HUGE savings.
Fully agreed with the conclusion. Staff compensation (including SUBSTANTIAL nonsalary compensation) is a big part of the budget of any school. Making the work of teachers more efficient (in the economics sense of "efficient") helps free up money for other worthy purposes.
Another comment to your comment doubted the figure, I think because you wrote "US government spent" when you perhaps meant "all levels of government in the United States spent" so much money on education. Most spending on schools in the United States is from state taxes (as in my state) or from local taxes (as in some other states), as I'm sure you already know but which may be news to onlookers from other countries.
"The U.S. has more than 14,000 public school districts and spends more than $500 billion on public elementary and secondary education each year (combined spending of federal, state, and local governments)."
(Billions of dollars. Detail may not add to total due to rounding)
Total Federal State Local
593.7 74.0 258.2 261.4"
Where did you get that $900 billion number?
However, I thought the aspects that was supposed to be "revolutionary" was detailed tracking and analysis of students and that feedback loop (either back into the system, to the student themselves or to teachers). They seem to completely ignore this.
Khan Academy isn't disruptive because of Salman Khan's lessons. It's disruptive because of the underlying platform, which any teacher (some possibly better than Salman) or any student could take advantage of.
All that to say: yes, a dude sketching some problems on a tablet is not the "revolutionary" part (if there is one at all) of Khan Academy.
I am looking forward to the future when a video lesson distributed to 20% of children on the Monday can have its feedback aggregated and analysed, with a patched version of the lesson pushed out on Tuesday.
For example, imagine teaching a maths lesson, and seeing that a significant number of children paused and rewatched a particular 20-second section. You could re-film that and splice it in ready for testing the next day. Imagine the possibilities if you had schools offset their holidays by a few weeks!
> teachers prepare mostly the same lessons every day
Centralized version will have none of that.
Of course if I'm dreaming that much I should point out that with each child able to learn at their own pace a single physical classroom could hold many different levels of the same subject at once, with children from different schools working together...
I find the AcademicEarth.org method of simply filming classes even worse. Yes, they do allow for what you'd normally call "quality education" to be accessible by anyone online, so that's a pretty big step, but I get this feeling that true disruptive online education is supposed to be something more radical...to make things 10x better for learning, not just replicate the offline learning experience.
What improvements do you hope to see? Making it easier to find short-term tutors would be on my list, throw up some IRC channels that tutors can monitor and get notified of students seeking help. I think one-on-one teaching is the most optimal.
As for how easy it is to learn, as long as we aren't hardwiring people's brains with information as seen in The Matrix learning is going to still take work. (Not necessarily hard work in every instance, but work.) I'd like to see more ways to interact and make the knowledge part of one's own experiences rather than as received truth (something Polanyi got mostly right in his Personal Knowledge book). Of course I'm instantly suspicious of any technique or product that tries to oversell its ability to teach concepts faster with better retention, and even more suspicious when it tries to sell itself as "it's so fun the user won't even realize they are learning!" That said, I can see benefits of various systems over both doing nothing and the status quo. Tools like Anki for instance work well, if used. The advice some programming texts give to manually type the programs also has obvious benefits, basically a subset of techniques that train the mind to follow and notice common actions. (There was an app linked on HN a few weeks ago that replicates the motions of solving algebra problems without actually doing any algebra, I can see how "eliminating two like things from both sides" feeling "okay" may help some people a little.) Khan has mentioned that he has offline students watch lecture videos as homework and do problems or answer questions during classroom time, the opposite of the traditional school, I can see its benefits over the traditional method too for classes where information is typically transferred via lecture.
What things like KA, AcademicEarth, and Wikipedia all have in common though is that they change the utility function of learning. That is, when they're not being used as crutches to get through the traditional schools. Students can learn what they want, when they want, stress-free. The classical model is "pass the tests with scores above this threshold or die." (Not literally of course.) It encourages cramming and forgetting. Cumulative testing doesn't help because the course only lasts maximally 9 months (with a big break in the middle), and at higher levels only 3-4 months. Sometimes it's even just one term rather than a full semester. It's not enough. Add in repetition of the wrong things (it's always the most basic things that get repeated endlessly when they're the easiest things to remember!), poor communication between professors of what they actually cover, poor communication between student and professor of what the student actually understands, fixed deadlines, anxiety over money (whether living expenses or paying $1500 for a dumb class), anxiety over spending X hours of your life for a useless class/assignment, retaking the entirety of a class instead of focusing on what you're missing specifically, being forced to take classes one has no interest or skill in, being exposed to one resource, et cetera, it's amazing the classical system works at all.
I like Khan Academy's videos. But if you watch Khan's TED talk, it's clear neither he nor bill gates sees khan academy as operating outside the education system. His lectures are meant as a complement to quality instruction in classroom by a teacher. Criticizing the Khan academy ecosystem as an independent and parallel instruction system is silly.
Khan is a great lecturer. He's better than every lecturer I've had. He may not be the best total educator. I wrote a blog post about this a while back: http://cyclicreality.com/post/26719449459/why-khan-academy-i...
The hype may be too much. Maybe it's not revolutionary. But getting a teacher who can explain and engage recorded and available free online is a great idea even if it's not the be all and end all of education.
I don't see the downside to such a great resource.
Also, there are many many terrible teachers out there. I had terrible teachers at the elementary and high school level. Teachers who were either incompetent or did not care. Khan Academy is not the problem.
As a recent graduate: Thank You, Sal! Keep up the good work.
I'm certain I read nothing more than snobbery in this article. It reads like "we didn't think of this free online disruptive academy first" or "how dare you do something differently from and outside of Official Academia."
Do you know who the best teachers are? I mean the absolute best, the ones who actually get what the student is struggling with? Other students-- the ones who just got through that particular lesson, the ones that just had the exact same struggle. The students' ability to teach like this declines with experience with the material because it all starts to become second-hand and they forget all the little things they struggled with early on. When an instructor takes note of students teaching students and the hows and whys of that interaction and remembers to cover the same territory in the next class, he becomes the best non-student teacher he can.
My mother is my perfect anecdotal example of this. In her 50s, she's getting a degree. While taking calculus, she also worked in the tutoring lab. After understanding new material, she'd be an absolute godsend for students for about two weeks and then her helpfulness would fall off. (You could also attribute this to all these classes being mostly in sync with at most two weeks of lag - that being the case, she would be less exposed to the reteaching of the material ... and forget those little details.)
One of the authors is a current teacher, working with real students. He gets feedback from them either directly (by questions, comments, curses) or indirectly (homework, tests, quizzes) to gauge his effectiveness as an educator. Khan (until they added problems to the site) didn't have this feedback.
Their criticism boils down to a few main points.
1. He uses an inconsistent presentation of algebra and arithmetic work and notation, which could lead to students becoming confused when he suddenly changes it.
2. He uses a poor selection of examples, without offering more or better examples students may develop a mathematical toolset based on a flawed understanding, which will lead to problems further down the line.
3. The problems available to students are insufficient to provide proper mathematical practice and ensure that some of the common misunderstandings are shown to be wrong.
There's a difference between constructive criticism that's effectively administered vs a publicly-rendered smear attempt that tries to hide behind pedantry.
Michael Paul Goldenberg has such a lengthy online trail of writings about mathematics education that as I entered his name in Google, autocomplete finished the search query
and I was led to some of his more recent writings. (I used to interact with him quite regularly online in specialized email lists about mathematics education reform, about a decade ago.) He, um, definitely has a point of view in his approach to education reform. It's all about the providers for altogether too many people who look at education results and school practices in the United States.
The response to the usual excuses for United States school performance by another observer,
who points out how often critiques of schools in the United States are responded to by excuses that shift blame from providers of "education" to the learners in their care:
"Consider that Americans tend to have more disposable income than citizens of other advanced market democracies, at least some of which can be devoted to supplemental instruction. After all, parents have fairly strong incentives to secure educational advantages for their children. This suggests that our schools are performing very poorly indeed.
"Don’t believe the hype."
is closer to reality than many of the critiques of outside-the-box approaches to mathematics education in the United States.
That said, I have been up-front here on HN in suggesting ways that Khan Academy can improve, for example by building more online practice that is truly problems rather than exercises (379 days ago),
"Just for friendly advice to the Khan Academy exercise developers, I'll repost my FAQ about the distinction between "exercises" and "problems" in mathematics education. It would be great to see more problems on the Khan Academy site."
and the Khan Academy developers have been listening, and I have had interesting off-forum email interaction with them as they attempt to improve the instructional model at Khan Academy.
To date, I recommend to my own children and to my clients in my own supplemental mathematics education program that they also turn to ALEKS
(Yet another edit. About the time I posted this, someone else asked below another comment,
So who is making the site that will deliver more personalized instruction? Where is the research that site will use, telling all about which kinds of personalization are proven and how much effect they will have?
and ALEKS is an answer to those questions in large part. Browse around the ALEKS site to see its links to its research base.)
and to Art of Problem Solving
for more online mathematics instruction resources, and I also share specific links to specialized sites on particular topics with clients and with my children. Besides that, I fill my house with books about mathematics, and circulate other books about mathematics frequently from various local libraries.
I also recommend that all my students use the American Mathematics Competition
materials and other mathematical contest materials as a reality check on how well they are learning mathematics.
In general, I think mathematics is much too important a subject to be single-sourced from any source. Especially, mathematics is much too important to be left to the United States public school system in its current condition.
I was just rereading The Teaching Gap: Best Ideas from the World's Teachers for Improving Education in the Classroom (1999) the other day. It reminded me of facts I had already learned from other sources, including living overseas for two three-year stays in east Asia.
"Readers who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about gaps in teachers' competence but about a gap in teaching methods." p. x
"When we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons--that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background." p. 77
Plenty of authors, including some who should be better known and mentioned more often by the co-authors of the article Colin kindly submitted here, have had plenty of thoughtful things to say about ways in which United States mathematical education could improve.
In February 2012, Annie Keeghan wrote a blog post, "Afraid of Your Child's Math Textbook? You Should Be,"
in which she described the current process publishers follow in the United States to produce new mathematics textbook. Low bids for writing, rushed deadlines, and no one with a strong mathematical background reviewing the books results in school textbooks that are not useful for learning mathematics. Moreover, although all new textbook series in the United States are likely to claim that they "expose" students to the Common Core standards, they are not usually designed carefully to develop mathematical understanding according to any set of standards.
In a January 2012 lecture,
Professor Hung-hsi Wu of UC Berkeley points out a problem of fraction addition from the federal National Assessment of Educational Progress (NAEP) survey project. On page 39 of his presentation handout (numbered in the .PDF of his lecture notes as page 38), he shows the fraction addition problem
12/13 + 7/8
for which eighth grade students were not even required to give a numerically exact answer, but only an estimate of the correct answer to the nearest natural number from five answer choices. Even at that, very few students chose the correct answer.
Patricia Clark Kenschaft, professor of mathematics at Montclair State University in New Jersey, reported in her article "Racial Equity Requires Teaching Elementary School Teachers More Mathematics" in the Notices of the American Mathematical Society
about elementary teachers' knowledge of mathematics in New Jersey:
"The teachers are eager and able to learn. I vividly remember one summer class when I taught why the multiplication algorithm works for two-digit numbers using base ten blocks. I have no difficulty doing this with third graders, but this particular class was all elementary school teachers. At the end of the half hour, one third-grade teacher raised her hand. 'Why wasn’t I told this secret before?' she demanded. It was one of those rare speechless moments for Pat Kenschaft. In the quiet that ensued, the teacher stood up.
"'Did you know this secret before?' she asked the person nearest her. She shook her head. 'Did you know this secret before?' the inquirer persisted, walking around the class. 'Did you know this secret before?' she kept asking. Everyone shook her or his head. She whirled around and looked at me with fury in her eyes. 'Why wasn’t I taught this before? I’ve been teaching third grade for thirty years. If I had been taught this thirty years ago, I could have been such a better teacher!!!'"
A discussion of the Common Core Standards in Mathematics, "The Common Core Math StandardsAre they a step forward or backward?"
gets into further details of how mathematicians look at the general school curriculum in the United States. It is not the worst curriculum possible, and survivors of the system often have access to outside resources to supplement school lessons, but the public school instruction in mathematics in the United States still shows plenty of room for improvement.
After edit: I was asked in a reply what I think about the essay "Lockhart's Lament." I think it is an interesting read, but less practical for reforming mathematics education than I had hoped. (I say the same in general about articles by Keith Devlin, the mathematician who popularized Lockhart's Lament.) To reform education, it is important to be relentlessly empirical, and look again and again and again at the best practices of the highest-achieving countries. That's why I prefer several of the links I submitted to Lockhart's interesting essay as policy guidance for United States parents, taxpayers, and learners.
Another edit: HN user danso just kindly posted
a link to a response by Sal Khan in the same Washington Post op-ed column about education. Direct link is
Since there is a strong, strong chance that many of the readers who were taught in a U.S. classroom never saw this, I have the above link included.
What would be meant, in the original article, by the "multiplication algorithm"?
I know that's a bizarre question, but actually I don't think I could explain what that means to someone myself, so I'd better ask.
for each digit in the bottom number, starting with the ones digit:
begin a new line beneath the current state.
write a number of trailing zeroes equal to the number of digits of the bottom number you've already processed.
for each digit in the top number, starting with the ones digit:
write (carry + top digit * bottom digit) % 10 to the left of the leftmost digit on the bottom line of the state.
the carry for the next digit of the top number will be (carry + top digit * bottom digit) / 10
use the addition algorithm to compute the sum of all the numbers you wrote this way.
And THERE is the difference between genuinely-given constructive criticism and the questionably-motivated smearing that those folks working with the Washington Post are attempting to deliver.
Thank you very much for the links. I've already sent some of them on to my wife so that she and I can research them a bit more as instructional aids for our children.
In fact it was very important to me that the piece I co-wrote <strong>not</strong> be a smear piece. I can't speak for the comments I did not write. But a careful read of the post itself should reveal a claim backed up by evidence. The claim is that Sal Khan is lacking some important knowledge for teaching. That's not a smear. I would gladly engage in a discussion that pointed to evidence that he does know some important things about how people learn.
The problem with education in this country is that we funnel most all of our public education dollars into a system and related organizations that are only tangentially motivated to educate our children. Watch how the public school systems and teachers unions react to vouchers and charter schools and you'll see how much they care about educating kids vs protecting their turf and power base.
I've noted that those within the education system or closely aligned with it absolutely hated "Waiting for Superman". They nitpick it and try to blow up minor faults with it in a similar way to how you're going after Khan. To those of us outside of that power structure, that documentary showed how the establishment is utterly opposed to actually solving problems in education in this country if those solutions occur outside of that establishment.
I view your articles vs Khan to be in a similar vein. What Khan has done is he's added a solution to our society for our education problems that operates outside of the current public education power structure. Rather than build on what he's done and improve on it through working with Khan or doing something better -- you folks have taken the route of trying to publicly take him down in the media.
His seems to advocate that children should be taught to explore math problems and make discoveries rather than memorize and practice given truths. My layperson's read of your comment finds some commonality between your positions.
Granted that there is a lot wrong with the current system, I'm curious what you think about the specific criticisms levelled against Khan in the OP's article. Is there any merit in the charge that he introduces the equality operator without enough explanation? Or that there ought to be more discussion in his lessons about the meaning of the decimal point?
"“I think frankly, the best way to do it is you put stuff out there and you see how people react to it; and we have exercises on our site too, so we see whether they’re able to see how they react to it anecdotally.”
Khan will put the video out there and see how people react to it. He perceives this to be a better approach than incorporating results of quality research projects into his instructional decisions. In the age of No Child Left Behind and its mandate for “scientifically based research” as the foundation for classroom instruction, this seems lazy."
I was agreed with the author until I read those lines. In my opinion they are disingenuous at the very least.
Every teacher, even those ones with a high knowledge on PCK, have to adapt their content every year. They try with an explanation on a topic and see how students react and based on that improve the content for the next year.
That's exactly what Khan said he is doing, publish and see how people react to those videos. It might be harder for him to measure the content in comparison with an on site teacher. But saying that "Sal Khan is on record as dismissing this research" seems an overstatement to me in the best case.
If you read the second to last paragraph, they don't suggest that Khan Academy shuts down, or that he stops teaching. They literally just say, "Hey, it'd be nice if he asked some really great teachers to look over his examples before he uses them to make sure he doesn't confuse people." I'm not an expert, but sounds like it couldn't hurt.
I view Kahn Academy as the start to a new type of education that is in its very early stages. Kahn is a smart man who will probably take the criticism and use it to be better his instruction and add supplemental videos etc. I would be shocked if he didn’t add in a 3rd comparing decimals video after this criticism.
Keep up the great work, I know I’m watching.
Just to point out this is the most ass backwards way to talk about decimal fractions I have ever witnessed, but nothing would stop said Valerie Strauss on pedagogical insufficiency spotting... "decimals for short — the numbers to the right of the decimal point" ?!? Whisky Tango Foxtrot
Edit: downvoted is it because you would also believe decimal fractions are decimals and those are the numbers to the right of the decimal point per article explanation... because being a silent ignorant with a karma finger really fits you.
"Decimals" is (in the US at least) a common way of referring to decimal fractions (that is, numbers represented like: 12.345) vs the whole number or integer representation that the students would already be familiar with (like 12) or other fractional notations like 6 7/10.
As for the abbreviation presented, it is just terrible and is used in article while accusing others of lack of Pedagogical Content Knowledge.
Her contribution is the first 2 paragraphs, the rest is the critique from Christopher Danielson and Michael Paul Goldenberg.
Edit: And please explain, how is the abbreviation terrible, because I've clearly missed something.
The abbreviation is terrible because in the article it is used in reference to "the numbers to the right of the decimal point"; "decimals for short".
To the second, I still don't see what makes that terrible. It's saying, decimals is short for decimal fractions, and provides a definition for decimal fractions.
Second: "Comparing decimals. Decimal fractions (decimals for short — the numbers to the right of the decimal point) are a notoriously challenging topic in the elementary math curriculum."
For the last time, decimals are NOT the numbers to the right of the decimal point like stated. "decimals" is being used for a subpar abbreviation of decimal fractions AND the left side numbers to the decimal point are a part of it, unlike what said authors state. Someone with pedagogical concerns should be awake when stating stuff like this and if you don't see the mess that is, that is your own problem. I have stated this already.
If someone comes up with a theory on a better way to teach something, cool. A/B test it and keep the winner.
The vast majority of contact hours in the vast majority of educational institutions consist of chalk-and-talk - someone writing things on a board and talking. The available evidence shows that there is no clear benefit to doing this in the flesh over delivering the same approach through video, and that there might be significant benefits to providing it in a modular format that can be paused and rewound. Sal might not be the greatest math teacher on earth, but soon enough, somewhere on the internet will be lessons by 99th percentile teachers on every imaginable topic.
If your best argument is "We can do what the internet does, only marginally better", then you're in deep trouble - we've seen how well that has played out for any number of people. The economies of scale are too great, the rate of iteration too rapid. You're just not going to beat the internet at supplying data. Educators and schools have to work out what they are uniquely equipped to do, or face the same inevitable obsolescence that is befalling libraries and travel agents and local newspapers and record stores and myriad other businesses.
(I'm of the belief that it's possible for a large news company to give equal commentary share to competing interests, but I don't follow the WaPo's education coverage enough to be able to make a judgment in this case)
This is sad because the actual message, which is valid in some instances and informative in all instances, gets lost in the noise. This is more so, given that people at Khan Academy are responding positively to specific concerns on specific topics.
And unlike for-profit companies, all his work, from the most boring video to the best, are available to be criticized. We only see a few clippings from, most likely, the very best videos from for-profit companies.
I hope the real critics and the real skeptics, i.e., those who would like to make Khan Academy a success, will continue to bring their specific concerns to the table; and just not all the distracting noise.
(Now, it is strange that the authors while pointing out the exaggerated praise for Khan Academy, fail to point out the abuse that Khan had to endure. Even stranger is that some commentators dismiss the abuse as "satire".)
I would think it would be easy for the "nit-picking experts" to produce that and it probably would have taken less time than the formal analysis they did. Probably cheaper too.
BUT the same effect is seen on Wikipedia, which has been reviewed and corrected more than thousands of other encyclopedias, simply because it's so open and accessible. Khan Academy was originally run by one guy, but they're starting to open up and partner with other educational entities. They aren't going to go unchecked. By no means is Khan Academy hurting the education system, unless you look at things from an economic or political standpoint, in which case it's potentially disruptive. But that would be cruel, wouldn't it?
I cannot disagree more. The best teachers I had were college professors who were all experts in their field and never took a class on pedagogy or education. But they had the autonomy to change their teaching style based on what was most effective for them and their students.
Our grade school teachers would be a lot better if they were experts in their domain of knowledge and given more leeway to teach according to their strengths and their students' needs. As it is, public school teachers are pretty much automatons whose actions are dictated down to the minute by school boards.
My worst teachers by a long shot were also content experts, holding, in one case, the research chair of an academic department in a renowned research university.
Couldn't teach to save his life and was lucky that the tutors put in a lot of extra hours so that around 50% of the enrolled students passed his fairly easy course (it was a revelation that after I had failed, and had to re-sit during the summer, I asked an academic in a related field to coach me based on a text, and his explanations were lucid, to the point, had an expository style that engaged and that he actually wanted to address the material at hand).
An intelligent person with some domain knowledge, good teaching materials and firm grasp on how students learn beats someone with all the domain knowledge in the world who is a hopeless teacher, in my experience.
If knowing math at a superficial level opens doors for other learning, it's hard to argue that it's NOT a worthy trade-off.
We as math teachers strive for perfection, but the truth is that having a deep understanding of what an equals sign is and how it should function isn't important for most people in their lives. Sure those in math careers will want that depth (and they'll get it, since Khan alone can't prepare them for that career), but everyone else will be fine whether they use the equals sign properly or not.
In a lot of ways it's like prescriptive vs. descriptive linguistics. Are we going to be the equivalent of a grammar nazi and come down hard on everyone we see that uses improper notation? Or can we be satisfied if there is some understanding there, even if it's imperfect or communicated poorly?
Game, set, match to Mr. Khan.
In my opinion Stack Overflow is a better model for this stuff. Imagine something like Stack Overflow focused on video tutorials covering concepts that got up voted. (heck, maybe it exists.) why can't I upload my take on "slope" (or whatever) rather than just post Q&A on Khan's?
This is ONE MAN.
"We contend that"
No. You cannot contend against 170 million.