Sal Khan responds to critic
washingtonpost.com
washingtonpost.com
I would like to correct a persistent misconception or two.
Persistent misconception: "...we suggest that Khan Academy desperately needs voices of teaching experience. Khan could tap into any number of existing networks..."
Truth: We have four ex-teachers as full-time employees. We have two high school math teachers as consultants. One Harvard Doctoral candidate in Education and one post-doc in neuroscience at Stanford are in residence. One UPenn Professor is also likely to begin a sabbatical with us. We have a 3 person team dedicated to working with and getting feedback from our 50 pilot classrooms and the 15,000 teachers actively using KA in classrooms.
Persistent misconception: "...it certainly requires more than just “two minutes of research on Google,” which is how Khan describes his own pre-lesson routine."
Truth: Go read Sal's AMA response (includes the sentence "When I did organic chemistry, I spent 2 weeks immersing myself in the subject before making the first video") before taking one of these "two minute" snipped quotes at face value: http://www.reddit.com/r/IAmA/comments/ntsco/i_am_salman_khan.... I've seen Sal's face light up when he gets an unwieldy new shipment of textbooks to start studying in preparation for his videos. Does he dive right into some videos? Absolutely. Is claiming that his "pre-lesson routine" can always be dismissed as two minutes of Googling disingenuous and patently false? Absolutely.
I personally think that Khan's response was one of tact and class, as is your comment. Keep up the good work.
I've tutored math in Korea, observed the schools in Japan and Singapore, was observing math class in Shanghai most recently, and have used curricular materials from Singapore, Japan, and China to homeschool my own kids in math.
What I observe in the Asian programs I've seen that differs starkly from the vastly inferior curriculum here in my own country (US) is meticulous care put in to the development of:
1) a strategically-ordered, cohesive sequence of topics,
2) explicit list of the ideas that need to be mastered in each topic before moving on, and
3) rigorously field-tested teaching techniques, examples, assignments, and assessments for each idea.
The whole sequence is constantly being experimented with, tested, and optimized like a Japanese industrial, or Apple consumer, product.
In contrast to this meticulously crafted approach to incremental mastery, our US approach is one of each teacher throwing her own bowl of math spaghetti at kids and whatever sticks sticks. Whatever doesn't stick today, well, don't worry, we'll be doing something different tomorrow. Maybe you'll do better with that. And someone else will throw the whole bowl at you again next year, and the year after that. If you run out of years and a lot of things still haven't stuck, well, we did all we could. You're just not a "mathy type" (claim the teachers who couldn't solve a Singaporean 5th grade word problem if their lives depended on it.)
What Sal has produced in version 1 reminds me more of the American way than the Asian way, but that's no surprise. It started off as an act of personal generosity, not a NASA space project. If it stays the way it is, I'll be grateful for the moral equivalent of free pizza. I'll gladly eat some, and I won't fuss that my free pizza doesn't have my favorite toppings.
But if he's planning to go forward, I think the next step might be to back off a bit on the quantity of videos and consider upgrading some of the old ones to a more deliberately designed, cohesive video sequence.
You can combine a well-designed, coherent curriculum for skill training without imposing the crushing baggage of "your worth as a human being depends on the brand prestige of the university you get into."
If you want to train people to be inventive in finding new ways to apply mathematics, and new mathematics to solve problems, then you need a process which leaves some room for questions, creativity, for open-ended challenges, and for some context around mathematics as a creative process.
The industrial production line approach may be efficient, but perhaps requires a lot of external pressure and discipline on kids to keep it on track, and can result in people who are strongly technically skilled within the relatively narrow boundaries of material they've been drilled on, but utterly lacking in imagination or passion for the subject.
In practise I suppose you need something in the middle. It takes disciplined study to develop the fluency to be creative in maths at a given level, but the discipline is a means to an end not an end in itself, and if it's so strong that people are discouraged by shame from asking the questions they need to ask to develop a deeper than rote understanding of the material, then it's got to be counter-productive.
I say this as someone who was almost put off maths for life by the latter approach, but is now doing a second masters in a mathematical field...
The points made about the specific videos may be valid, but those points about specific videos and ideas, can be put across more effectively if the authors get these facts correct.
I'm sorry, but 2 weeks is not that much time. Most TA's for ochem classes have at least taken 2 quarters (and yes, I can assure you, they also are 'immersed') of the subject, and they aren't in charge of the lecture.
Honest question. I have not seen them and I am not in a position to evaluate them even if I had.
And look at how many still blow up or otherwise fail to put their payload in the desired orbit (although maybe in general we're getting better at that).
You can have as many theories as you want on this matter; I have many of my own. However I'm asking if anyone in a position to actually evaluate what he already did can tell us how well it worked.
A Usually Persistent Truth: the farther you are from when you learned something, the worse you are at teaching it. Khan's ability to learn something and turn around and teach it is likely the reason it's accessible and enjoyable for students.
I haven't viewed any videos from Khan academy, but I think a public repository of lectures that are rigorously checked for correctness can only be a good thing. Is that what Khan's academy really is though? Teachers are trained to engage young minds. Is it realistic to expect flawlessly correct lessons from someone with this background after just two weeks of immersion in a given topic, especially when reference materials (e.g. highschool textbooks) are so often riddled with mistakes? If you're teaching math or physics, why not collaborate with a mathematician or physicist? Let Khan's teachers handle how to structure lessons and provide engagement while experts in their field provide the material. There are plenty of experts out there who would jump at the chance to help, if only to save themselves from headaches when teaching first-year courses!
I also think people often overlook the fact that Khan Academy isn't really meant to be a revolution for the worlds privileged students. What KA really does is provide access to learning materials to people all over the world who do not have access to good teachers. People like to point out that recorded lectures do not provide the interactive element that real teachers do, and that's absolutely true. However, there is no substitute for homework. To learn math, one must do math. No amount of interactiveness can change that. If KA provides good exercises along with its lectures then it could indeed provide disadvantaged (but motivated) students with a road to academic success anywhere in the world.
Anyway, I agree with you on KA. My big thing is, and I think Khan is an awesome, amazing, but when we get down to brass tacks does KA help students really "get it done?" I don't know.
I feel like there's always all this talk about Khan Academy and how great it is and all of this. But if you get down to it and watch the videos for hours, not just talk about watching them, how much is it helping? Maybe it is but I've showed it to a lot of people and have tried, and this is just my anecdote, to get people to learn via KA and their results haven't been amazing or anything.
But there's no reason it can't improve and get better. I just get this feeling that we have all collective gone "OMGZ MATH VIDEOS!!!!!!!" and haven't done a critical examination.
That glass is liquid and old panes of it are thicker on the bottom because it flowed. (The definition of a 'solid' is a bit technical, but panes of glass do not flow.)
That raindrops have a 'raindrop' shape. (No. Wrong. Not possible to salvage this one, as it is entirely incorrect.)
> I was made aware of the fallacy of the glass flows myth many years ago by the late great glass chemist, Nick Labino. Nick offered this simple analogy, "...if the windows found in early Colonial American homes were thicker at the bottom than the top because of "flow" then the glass found in Egyptian Tombs should be a puddle."
http://www.cmog.org/article/does-glass-flow
> Many years ago, Dr. Chuck Kurkjian told me that an acquaintance of his had estimated how fast—actually, how slowly—glasses would flow. The calculation showed that if a plate of glass a meter tall and a centimeter thick was placed in an upright position at room temperature, the time required for the glass to flow down so as to thicken 10 angstrom units at the bottom (a change the size of only a few atoms) would theoretically be about the same as the age of the universe: close to ten billion years.
I can find more if you want.
Edited to add: Oh what the Hell:
http://math.ucr.edu/home/baez/physics/General/Glass/glass.ht...
http://www.thefoa.org/tech/glass.htm
(The FOA is the Fiber Optic Association. It seems every serious group that works with glass has debunked the flow myth.)
I guess my point was that it's just as likely for a teacher to accidentally feed you misinformation as it is for KA. More likely even, given that KA is transparent and open for scrutinization.
There will be many teachers who will use Khan's videos in their teaching (or to augment it), and I imagine that over time Khan will change the way he does things based on his own education about education.
It's self evident that the 'sitting in front of a machine watching videos' isn't the solution to the education. If it were the multimedia revolution wouldn't have petered out as it did. Children (and adults) need a variety of approaches. Khan's is just one.
Sal Khan(it' khan, not kahn) himself mentioned he is not looking to replace classrooms. He wants to invert the classroom. Traditional classroom has passive lectures followed up with problem solving which is to be done at home. He says it's better if people can do the passive things at home at their own pace, and use the classroom for problem solving.
The problem with passive lectures in classroom is many a times students are sitting through lectures they don't follow because they don't know the prerequisites; and often times students by themselves aren't able to figure out if they can't make heads and tails of probability problems, what is it that they need to know so that they can understand it. In his talk, Khan demonstrated the software(I haven't used; I might be off) which can track your deficiencies. For example, you are struggling at basic probability. The framework will drop you down to permutation/combination problems. You still are struggling; introduce fundamental principle of counting. You are doing fine now. So explain permutations based on counting principle.
Lacking the prerequisites is one of the problems. We have people who learn at different rate, people who aren't fluent in the language the lecture is delivered in, people who are shy/under-confident to speak up, people who are smart and are bored and feel left out, people who aren't very smart(lack the fundamentals, slow learners...) and are lost and feel left out.
If Khan Academy, or any other self-paced learning resource can solve these problems; coupled with classroom guided problem solving and projects, that will make a huge difference.
I do feel that those times have changed dramatically. Now all we hear about is the college dropouts who went on make billion dollar companies. It has almost become cool to not go to school. I think the time is finally right for these non-traditional educational services and it just helps that the internet is now widely available to act as the conduit for it.
When I went to high school in Pennsylvania the 90s people were still encouraged to into the trades. I have some friends from high school that have gone on to be very successful electricians and welders. Even when my brother went to the same school and graduated in 2007 he was never pushed to go to college. He ended up getting his AA at a local university and is now a paramedic/fire fighter for the same city.
If you think the folks commenting here have any idea what life outside the bubble is like, think again.
I graduated in 2002, at the time any Georgia high school graduate with a B average got full tuition and fees paid for with the HOPE scholarship (still exists, but it only pays about 80% now).
Possibly as a result, almost everyone who wasn't a terrible student was encouraged to go to college.
The only people pushed into the trade direction were people with less than a B average, and with grade inflation you really had to be pretty lazy not to get by with a B.
These billionaire dropouts like Zuckerberg and Gates, ... where did they drop out from? Although I don't think it makes a lot of sense to equivocate those two, it will suffice for now.
I don't think times have changed that much. Learning purely from these things is also going to be likely only for autodidacts. There happen to be many of autodidacts in our little world (or at least autodidactic enough to put together a website in Rails or something) but this is not true of that many people.
In fact, how far do you get with this online stuff? Not far, if you look at most places like Khan and so forth, you're talking about sophomore, maybe junior year stuff for the most part. OK, sure there's the occassional machine learning course that gets popular online that a lot of people that don't have any sort of mathematical background take and can say they are a part of the "cool kids" group, but let's be honest. Even then, we're still talking about this small segment of autodidactic or semi-autodidactic types.
A lot gets said of "big bad college" and how the "fresh new hip information revolution trumps traditional learning" but I don't see it. Sure, you could put up a few tutorials about learning to make Hello World in Javascript but I'm just not seeing this education revolution here.
With people like Zuckerberg, Jobs, and Gates frequently in the news as of late, people are starting to question those long-held beliefs. They see that people are successful without the education, despite what they've learned to not be true. The stigma of not going to college is starting to lift as a result. Knowing those famed people personally is completely irrelevant here.
If spending a couple of months reading up on Rails will get you a decent job, that's exactly what people will do. They really do not care about deep CS topics. They care about providing a decent home and comfortable living conditions. That is it.
I respect the academics, but you are as rare as the autodidacts.
What a dumb thing to argue about. I'm not a historian (or a mathematician), but the term "slope" seems pretty obviously adopted from a physical slope/incline/hill. Why? Because it's the easiest visual analogy for us apes to grok. It doesn't come from an earlier term meaning strictly "a rate that describes how two variables change in relation to one another."
If you're trying to teach someone a complex concept, are you going to use a phrasing that has zero significance to them? Rate? Variable? What? The people learning about slope aren't programmers or engineers. Give me a break.
Why not use a visual analogy that makes perfect sense and is still a valid definition: rise over run on a section of a hill, road, roller coaster, etc. I hope whoever is teaching this is relating it to a real world object. Just talking about a line on paper isn't going to help much, but neither is an overly complex definition.
>Take Khan’s explanation of slope, which he defines as “rise over run.” An effective math teacher will point out that “rise over run” isn’t the definition of slope at all but merely a way to calculate it. In fact, slope is a rate that describes how two variables change in relation to one another: how a car’s distance changes over time (miles per additional hour); how the price of an iPod changes as you buy more memory (dollars per additional gigabyte).
Followed by this in her response to his letter:
>As math was not my subject in school, I don't know who is right but would love to hear from mathematicians out there.
I had to reread all 3 articles multiple times. Then I lamented the sad state of journalism as well as the complete willingness of our society to tolerate the "I'm just not good at math so I won't bother to understand it because I don't think it's worthy"-attitude. What angers me about the first article is not that it went by some editor without the editor saying, hey, this seems suspect, but rather the fact that the journalist decided to write about something she had little knowledge about with a tone that suggests that she thinks she knows knows more than Sal or someone else educated on the subject. An unapologetic combination of ignorance and condescension.
On a related subject; note to WP: do not have guest posts published under an editor's byline. Instead of
"By [editor]
[italicized paragraph explaining that this is a guest post published by an editor]
By [actual author]"
Have 1 byline... the person who wrote the article.
> Below is Khan’s e-mail to me, which I shared with the author of Monday’s post, Karim Kai Ani, a former middle school teacher and math coach who is the founder of a company called Mathalicious. He said Khan is wrong.
So, to summarize, "Khan is wrong, but I won't bother to explain why, he just is, and I have a self-proclaimed expert that says so."
There were some cogent points in Kai Ani's critique; the biggest one I think Khan would completely agree with: education problems aren't going to go away because of videos on YouTube.
Moot point since the Khan Academy is much more than about YouTube videos now.
The videos may have kickstarted it; but to offer the above as a critic to Khan just shows that you are debating on an outdated version of what the Khan Academy is up to.
http://mathworld.wolfram.com/Slope.html
http://en.wikipedia.org/wiki/Slope
I cannot stand people who leave things at "just because". I can't remember who said it, but it goes "people who have explanations will explain".
I think what you mean is that a two variable linear equation over the reals describes a line in the Cartesian plane.
Try thinking of a line's slope (as you've described it) as just one measure of that line's "steepness" in the Cartesian plane. Thinking like this, we could use any number of measures to describe "steepness" (for example, angle of inclination provides a measure for the "steepness" of any line)
So it's sort of a philosophical difference. However, there is a a very subtle elegance at work here.... we have this geometric concept of "steepness" which -- very interestingly -- turns out to be related an algebraic concept.
Actually, we can go further. The geometric concept of "steepness" is related to algebra via a simple quotient, which is further related to an analytic concept (rate of change).
It seems so trivial, but a good number of mathematicians make a living by finding small connections between geometry, algebra, analysis, etc.
(Edited)
I don't know if she has a position on Khan herself.
"Below is Khan’s e-mail to me, which I shared with the author of Monday’s post, Karim Kai Ani, a former middle school teacher and math coach who is the founder of a company called Mathalicious. He said Khan is wrong."
"Below is Khan’s e-mail to me, which I shared with the author of Monday’s post, Karim Kai Ani, a former middle school teacher and math coach who is the founder of a company called Mathalicious. He said Khan is wrong. This won’t be the end of the debate."
It was part of the recap, I understood it to mean "Khan responds to this guy who posted the other day who said Khan was wrong" and then "stay tuned for his response to Khan's response" rather than "And then he replies Khan is wrong in his response. Stay tuned."
"After earning three degrees from the Massachusetts Institute of Technology (a BS in mathematics, a BS in electrical engineering and computer science, and an MS in electrical engineering and computer science), he pursued an MBA from Harvard Business School"
One would hope that his education background might have given him some insight into what "slope" was, and how to best explain it to someone in the 8th/9th grade.
I can _recall_ when it was introduced to me, and I didn't really grok it until a few years later - "rise over run = slope" was complex enough for my brain back then - I can just imagine if people started yacking about "Ratios of variables" - my head would have exploded.
The proof should be in the level of understanding of the concept of slope acquired by students using various methods; it doesn't matter at all who is "right" with respect to some formal definition.
Now, with that said - there _is_ a danger in that model of teaching, in which the student gets lured into a zone of comfort. So there _absolutely_ is a place in teaching for instructors who are going to challenge, upset, and disturb the student - resulting in a form of stress that pushes to the student to new heights. This can be a very uncomfortable (and, indeed, upsetting/stressful) learning environment - but it does give students deeper, and sometimes much more meaningful insight into topics.
But that's not what Khan's about. He is the guy you go to when you just want to get over some hurdle about a topic that has frustrated you.
> An effective math teacher will point out that “rise over run” isn’t the definition of slope at all but merely a way to calculate it.
Depends on the level. For introductory algebra, or classes before calculus, where the "slope" concept is used exclusively with linear functions, "rise over run" is a perfectly adequate definition.
For calculus, you can rigorously define slope as the limit of (f(x+h) - f(x)) / h as h -> 0 (or any of several other equivalent definitions). The subtleties of the proper definition are likely to be more confusing than enlightening to a beginner.
But even the rigorous definition is still a "rise" term f(x+h) - f(x) divided by a "run" term h. The phrase "rise over run" is an easy-to-remember mnemonic which actually does a really good job of capturing the underlying idea without going deep into more technical issues involving limits (calculus) or the guts of exactly what a real number is (real analysis).
Once the linear case is completely understood and the student has some basic facility with both mathematical reasoning and algebraic manipulation, THEN is the time to introduce the more general definition of "slope". Even then, mathematics teachers tend to use the word "slope" mainly in the linear case, and "derivative" in the general case.
As far as I know, this has been the standard way to teach high school / early college level mathematics for decades.
That leap you made there at the end of your sentence is the crux of the issue. Most educators argue that being an expert in a subject is not even remotely sufficient qualification to teach it effectively, especially to children.
Go find a professor, you sorry excuse for a "journalist".
Anyways, edited.
Basically, the traditional schooling model consists of "lectures" in class, where the teacher presents you with information for an hour, and homework, where you work on problems relating to that content alone in your own time.
Khan says, that's cocked up-- we should let students consume the raw informational content on their own time, where they can pause, rewind, and go over it as many times as they need to in order to understand it without disrupting anyone else, and then do "homework" in the classroom, where there's a focused environment that encourages exploration and somebody who can help each student with their individual difficulties.
Which has always struck me as a pretty straightforward, good-common-sense approach to at least try. Why is this concept so opaque to so many people?
The Khan and homeschool model, in contrast is the mastery model. Students are individually taught at their own pace. They are not graded and pushed on. Instead they keep learning a topic until they know it. In traditional classroom education, the learning is variable while the time to learn is constant. Khan makes the learning constant and varies the time. Some students will learn faster than others, but all students learn to a high standard.
Khan is not just delivering a better classroom. He is offering a wholly different model of education.
My issue with home schooling is the lack of variety and specialization. With only one teacher, the child just isn't going to be exposed to as many viewpoints as they would in a traditional school setting.
Homeschooling is still regulated by the state. Students must still pass certain tests, and often homeschoolers have a community of parents and children who get together to learn with each other at educational events geared specifically to their education. Both models have evolved, though they still have some cons. So, too, do public and private schools. Personally, I'd like to see us picking the best parts of various educational systems and putting them together.
Desire to provide religious or moral instruction is at the TOP of the list of reasons people homeschool.
Here is the survey's remark on their estimation methodology:
"When applied to survey data, weights allow for the generation of national estimates from a sample of respondents. They also adjust for characteristics of the survey design, nonresponse, and noncoverage. However, biases may exist in the data if weighting procedures have not adequately adjusted for these issues. A large-scale bias study was conducted in conjunction with the 2007 data collection. Readers interested in the findings of the bias study, as well as detailed information on NHES survey methods, weighting, and response rates, can refer to the Data File User’s Manuals published online at http://nces.ed.gov/nhes ."
I'd expect a measure of homeschooling families to want nothing to do with the federal department of education.
edit: (actually... It wasn't a heavier focus on science & tech... it was just more intense in general... I had a hefty literature portion (I read many Dickens books...) and some sweet art history books)
The real problem is the media. And by that overarching term, I mean the rhetoric that various journalists, bloggers, and others have let themselves use for whatever purpose (ie. sensationalism, pageviews, linkbait, etc).
It's understandable that there's been a backlash to Khan. He got overhyped. The pendulum swung too much one way. Now it's naturally swinging the other way.
In the end, this is going to turn out better for students. As critics lash out in both directions (supporters and detractors both have gotten pretty vicious in this debate), there are a bunch of for-profit and non-profit efforts that are creating alternatives. Khan has a smart team too. I'm sure they're steadily improving their offerings.
[1] http://www.washingtonpost.com/blogs/answer-sheet/post/khan-a...
He's addressing an EXAMPLE the critics used to illustrate their actual point, and fails to completely address the bigger issue.
I thought the critics' points were completely valid, and raise pretty serious issues about the quality and usefulness of the Khan Academy material.
The thing that annoys me about Karim's argument is 1. There is no constructive criticism. He does not tell Sal (or me the reader) how one can make SA better 2. He has a competing for-profit venture that is a direct conflict of interest with SA and leads me to question his motives 3. He is a self-proclaimed expert with no proof of his own expertise 4. From his rant, I don't get that there is something structurally wrong with SA videos in general as his claims make them out to be
"Quality and Usefulness of Khan Academy Material?" - you must have gone to an incredible school if you find his material lacking.
I didn't say anything about how I view Khan Academy content.
I don't use Khan Academy, because I have no use for it. That doesn't, however, preclude me from assessing critique made by others about it.
I have seen over 100 of Khan's videos. On many occasions he was able to clearly communicate a topic that I had been gated on. Doing so in a clear, concise manner. I have colleagues at work that use Khan to help their children through Math. If Khan had existed when I was in university, it would have eliminated 90% of the tutors costs associated with things like linear algebra. So, as I read his criticism, I was asking myself, WTF - this feels like it has _nothing_ to do with the content that I've viewed, which has been _excellent_ in explaining topics that very few (if any ) of the lecturers I had in High School/University were capable of doing.
What it actually reminded me, was of Encyclopedia Britanica's FUD against Wikipedia when it was coming out. The criticism _sounds_ accurate, unless you've actually looked at the content being attacked.
My point is - you can't read the criticism, and make a judgement, until you've spent some time looking at the material to see if the criticisms are sound.
From what I understand, Khan keeps the videos unscripted so he can maintain more of a conversational style, but he's always careful to understand the material very well before doing a video on a subject.
One thing I found quite frustrating was the lack of technology applied to making the video - in the few I've seen he takes ages drawing out number lines (for example) and the writing is quite unclear; it doesn't appear to be too hard to have a marked axis that you can paste in when it's needed. Petty issue but something I didn't expect to find on this much touted resource. Another example would be hand drawing bunches of marbles/tins http://www.youtube.com/watch?feature=player_embedded&v=D...; I guess it naturally limits the pace, which may be good.
Classroom teachers make mistakes. Textbooks make mistakes. But the system is set up to disallow question of these two authorities. Example: Math class is the first class of the day; teacher follows textbook and textbook is wrong; child questions and is waved off; is that kid gonna remember at the end of the day that she needed to ask mom about this problem? No.
I much prefer when Sal makes a mistake because it makes for a learning experience. What's nice is when my son comes back some time later and tells me that Sal's video has been corrected.
A good teacher can handle the book being wrong. A good teacher will tell the students that the book has a mistakes on this problem well before they have the chance to discover it themselves. A good teacher will have gone over the material in advance and know what's correct. Instead, we have teachers thrust into roles that aren't near their specialty; teachers that can get a degree and qualify for a paycheck because they can work the system, not because they know how to teach students. And interfering with that teacher's conveniences is not a good thing for a student to attempt.
1) Post a controversial guest article bashing a guy whose actually trying to do something good in the world.
2) Act like the neutral party so they don't have to take any blame. Allow Karim to be the scapegoat.
3) Sit back and enjoy while traffic explodes to their site.
I see you Washington Post. You ain't fooling me!
(Cross posted from the original thread because I genuinely would like an opinion more informed than my own and the totally unhelpful "conclusion" provided by Valerie Strauss.)
*Edit, I missed that he (Khan) posted a video defending his definition http://www.youtube.com/watch?v=TNaQJjLAhkI
Once you get to calculus, things get a little more complicated, as the slope at a given point is defined using derivatives, but I'm pretty sure that ends up being the same as the rise over the run of a tangent line.
In any case, anyone who would make such a lame criticism, should just STFU. If that's the best criticism they can come up with, they surely can't have much of value to say. Also, when teaching something like algebra, it's more important to make the material approachable and comprehensible, rather than define everything to a level of rigor that would make Russell and Whitehead happy.
In that case, this debate really isn't about you. You were probably an exceptional student, who saw the connections between mathematical concepts easily, regardless of instruction. You probably found yourself predicting the next concept a teacher would introduce, because it just "makes sense." Not all students are that way. Most are not.
The two people involved here are fighting over two different ideas. Sal is being pedantic, but is right, slope is defined as ∆x/∆y. What the other guy was saying is that slope represents rate of change, which is a much more important concept to early algebra, and the underpinning of why you actually care about slope in physics and calculus. You probably made the connection effortlessly. I assure you, many students do not.
I teach high school mathematics to both honors and special needs students, and it's important to keep in mind that the instruction is very different between the two populations.
Do you mean ∆y/∆x?
If so, that's a completely different criticism, however, from the criticism that Khan is putatively making an alarmingly dense stream of gross factual errors.
I think that we can all agree that Kahn is not the best possible teacher that exists in the world for each given subject. Is that a decent argument against what he has done? Hardly! That would be letting the perfect be the enemy of the good.
Considering that so many people learn from the Kahn Academy these days, an argument can certainly be made that Kahn's lectures should all eventually be replaced with lectures by the actual best teacher in the world for that given topic. For all we know, this is already in the works.
You may as well say it represents a tangent. (Pun not entirely intended.)
1. You could argue that if you're talking about slope, you mean slope of a line. If you want to talk about the generalized notion of slope of a line, you should use terms like gradient, derivative, etc. If Khan had taken this stance, I would have been fine with his defense.
2. You go with the fully rigorous definition of slope as basically being a synonym for gradient. This is what mathworld actually does. This is where Khan's reputation gets really knocked, in my opinion. He quotes mathworld (http://mathworld.wolfram.com/Slope.html) but only selectively. What he failed to mention are these key points:
- The very first sentence defines slope this way: "A quantity which gives the inclination of a curve or line with respect to another curve or line." - The sentence he did quote begins with "For a line in the xy plane..."
I am perfectly fine with Khan teaching it as "rise/run". I thought criticizing this was silly on Karim's part. But then Khan was the one who came back to argue that his definition was correct and Karim's was wrong. I think Khan's wrong here and shouldn't have even engaged in such a trivial dispute...
Khan saying that Karim was wrong, is wrong and actually made him look worse in my eyes. In the video he said "Slope can present rate of change." No, slope is the rate of change. Him saying Karim was wrong about the price vs gigabyte being the inverse if you switch the axes is ridiculous. By definition, you always say the first variable vs the second variable, where the first variable is the y axis and the second is the x. Khan should have left it at that instead of trying to twist things around to make Karim look wrong, it was a poor and transparent attempt at being vindictive.
Most definition use a phrase like "such that", which says nothing at all about calculating.
"Rate" implies that the denominator is a timespan. This does not make sense, as there are many slopes which are not rates and which are not presented as rates.
It's kind of a silly thing to worry about, but Khan's answer is unequivocally better.
To take the example of price of an iPod vs. more memory. The slope is inverse depending on which variable you choose to plot on which axis. If you just watched Sal's one video, you would not understand why that is... your only context is 'rise over run'. If given sample data points and asked for a slope, you might not be able to figure out which way to plot it based on the question.
To be fair to Sal though, this was just a single video subtitled "Figuring out the slope of a line" and so Sal may have covered the larger concept and its importance in a different video.
Approach 1, Mathalicious: "Unless you give students the right information from the first, even if it is a bit more abstract, they won't be prepared later on."
Approach 2, Kahn: "The best way to prepare students is to keep things simple. Later you can give them refinements about tangents and derivatives and such."
This is an interesting point, and I'd be pretty surprised if there isn't already a wealth of knowledge surrounding it in educational research. (Neither Kahn nor Ani appeals to such research in the articles.)
Of course, the slope of the position curve is the velocity; what I'm suggesting is that referring to it as slope detracts from understanding, while referring to it as velocity enhances understanding. Consequently, getting all wrapped around the axle over the canonical definition of slope is a shining example of majoring in the minors.
But when that relationship isn’t so clearly implied, sensible people don’t try to describe it by calling it a “slope.” Instead, they say what relationship they really care about: Is it the instantaneous rate of change at a specific point? Or at all points along a curve? (Or surface?) Or is it the average rate of change over some interval? Or the weighted average rate of change over some interval of varying density? Or the weighted average rate of change over constant-width intervals centered at certain (or all) points along a curve? Or is it really that they care about—?
You get the drift: If you care about rates of change, you’ll use the term “slope” to describe them only when the context implies a clear “rise over run” relationship between the variables of interest. Kahn seems to get this; Karim Kai Ani, not so much.
Among others.
Most public school teachers are awful. They have degrees in education, and most university programs in the teacher pipeline are intentionally easier and less demanding than the equivalent 'real' degree in a given subject. 'Math-ed' is a very undemanding degree compared to 'Mathematics'. If high-school teachers were placed under this kind of scrutiny we would be forced to completely re-evaluate how teachers are credentialed and licensed. They aren't put under this or really any scrutiny at all, and probably they never will be. Teacher unions fight as hard as they possibly can to prevent any measurement or evaluation of teacher performance.
Students will continue to try to fill in the gaps in their understanding of these topics, and if there are free resources available to them to do so, even better!
I've had dozens of amazing teachers throughout my education, all of which were excellent at gripping my attention, having a passion for their subject, and a knack for explaining it extremely well. If I thought my teachers were this good, can you imagine how good the best in the country would be? And what it could do for education to makes those lessons available for the entire world?
It's a shame he does it entirely himself. It's not for lack of funding, that's for sure. Maybe it's an ego thing?
The hardest challenge is this: disrupting the educational experience requires buy-in from teachers, administrators, regulators, academics, and existing service providers, but all these parties are nearly always resistant to change and well-entrenched in their positions, making large-scale change very, very, very slow and difficult.
The optimist in me wishes Sal Khan and his team only success as they take on, and attempt to co-opt, the educational establishment. The realist in me thinks they face a long, tough battle.
Aside from that, the only way the "(highly unionized) teachers do it better" argument can hold water is if the (highly unionized) teachers start producing students that actually place in a reasonable range in international comparisons.
These incentives should frame everything we hear about this exchange.
that conveniently forgets to mention all the collaborative/interactive components built into coursera - the community TA's, the forums, chatrooms, etc etc etc.
moral: don't criticize unless you've actually read the book/seen the movie/etc. or maybe beware of hidden agendas.