Alternate title: You Khan't Divide a Vector by a Vector.
Get it? Maybe you don't, but that is ok.
Really, I was going to make this a video critique of Sal Khan's – video critique of my video critique. If I made this a video, I could use this meme.
But that would be wrong to make this a video just so I could use that meme, right? I really was torn on this issue. However, I feel most comfortable with words and pictures and not videos.
Oh, I know what you are saying. You think I should just let this whole Khanpocalypse pass over. You are like Dr. Henry Jones: "Indiana, let it go."
I should listen to you or I could fall into the endless abyss of youtube comments. That would be bad. I never listen, do I?
Vectors
Let me be constructive. This is a really a good time to talk about vectors. Oh, and I should note that I probably came off as "rude" in my first video critique because I was going for the Mystery Science Theater 3000. That was part of the Khantest.
The central discussion seems to be over what the definition of a vector is. Khan says it is something with magnitude and direction. That isn't a terrible definition. Really, most textbooks say the same thing. Personally, I like this definition:
If the vector represents a real three dimensional (or 2d) thing, you could also say it has magnitude and direction. But I also like the Wikipedia definition.
Vectors in this vector space (which is really fun to say because you sound cool) have the following properties:
- A vector plus a vector is also a vector.
- A scalar value multiplied a vector is a vector.
- The cross product (vector product) for two vectors is also a vector.
There are many other properties of the vector space, but those are the important ones.
Can a Vector be Equivalent to a Scalar?
Really, this is the big problem I mentioned in the Khan video (impact velocity from a given height). Here is an example of his mistake.
I see my students make this mistake all the time - which is why it is important to NOT make the same mistake. This says that the variable on the left (the vector velocity) IS THE SAME as (or you might say "equal to") the thing on the right. Khan puts a scalar "zero" on the right. But scalars aren't even in the same space as 2D or 3D vectors. They can't be the same. It's impossible, even for a computer.
Oh, I am just being nit-picky. Yes, sort of. Students will say "what's the big deal. If you call it a vector or a scalar, you get the same answer". Well, I will show you in a little bit a case where you can get yourself into trouble.
How about another example? My kids play soccer. When they are young, they just want to get the ball in the goal. Often they will kick the ball with the front of their toe and the ball goes in the goal. No big deal, right? Well, for now doing this the wrong way worked out ok. However, if they keep kicking with their toes, they are going to have a bad time.
So, how would you say the ball starts from rest as a vector? You could write it like this:
All I did was to put the vector symbol over the zero. This makes it "the zero vector". In three dimensions it has the components 0 m/s, 0 m/s, 0 m/s. It is not a scalar.
How Do You Describe a Vector?
I actually have a very very very detailed post on the representation of vectors. Instead of going over all that again, let's just look at what Khan says. He says that there are two ways to talk about the velocity of the object. (this is from his – video "correction" of my video)
Here he claims that the "up" means it is a vector quantity. I am ok with that. What if you have a 1-Dimensional vector? How would you represent the one component of this vector? If it was "up" it would be a positive component and "down" would be a negative. The point is that Khan is saying that 1D vectors are very different than a scalar. That isn't true. If scalars can be positive or negative then a scalar is just a 1D vector.
This seems to be the main focus of Khan's rebuttal of my critique - but he missed the bigger problems. The other issue he addresses is that I say this is a 1-D kinematics problem and doesn't require vector notation. That is correct - but if you want to call a scalar value a 1D vector, then fine.
What About Negative Vectors?
This was a much bigger issue than the "up" vs. "down" vectors. Khan states that it is important to realize that the ball will be moving down and thus it has a negative velocity vector. This is another mistake I see introductory students make. If I were to write the velocity of a ball moving down as a vector, it could be like this:
(I wrote it two different ways) Just because one of the components is negative, that doesn't make the whole vector negative. Now, if you want to multiply the whole vector by the scalar value of -1, you can.
If you follow the normal conventions for vectors, putting a negative sign in front of the vector might do the opposite of what you want. Again, some will say "I know what he meant....stop being picky". We have to have a convention for representing vectors that is clear. I guess this would be the same as using short-hand "text message" style words instead of more formal words. They might get the point across, but I wouldn't encourage that in a writing class.
Vector Division
Check out this part of the Khan lecture.
How do you divide the change in velocity vector by the acceleration vector? Who knows. I guess it depends on how you define vector multiplication. What is division anyway? I would say division is the inverse operation of multiplication. The problem with vectors is that there is no plain multiplication. Like I said before, you can do the scalar product operation or the vector product operation but neither of these two leads to a nice division operation.
Really, this just shows that Khan is not appropriately using the tool known as vectors. If vectors were a lawn mower, Khan would use it chop lettuce.
Very, very, very, very wrong?
My claim was that in order to find the speed of an dropped object, you don't need vectors. Sal Khan claims that this part of my video critique is very times 4 wrong (I counted 4 very's). Was I wrong? Sadly, no. I stand by my statement. You don't NEED to use vectors. Can you use vectors? Yes. Can you use calculus? Yes. Could you model this with a real object and a stop watch and drop it 100 times to make a plot of time vs. different heights and then plot that to find the slope and velocity? Yes. Could you use solve this using elephants? Yes. Do you NEED to use elephants? No. Have I made my point? Probably not. Do I ever stop? Clearly not. Am I out of control? Isn't it obvious?
This is really the sad part of Khan's rebuttal. He picked the first thing I said and focused on that. I would love to hear him explain how to divide two vectors.
I need to use memes more often, they are fun. Oh, I can recommend mememgenerator.net - quick and easy.
One more thing about being wrong. I probably over stated Khan's wrongness in my original critique. Really, that was mostly for effect. Perhaps a better thing to say was "that's not the best way to do the problem". Doing it Khan's way can lead to problems with students later on. I wouldn't say he is very, very, very, very, wrong. Not even very wrong.
A Vector Example
How do you learn the difference between a scalar and a vector? Khan's BFF Frank Noschese has a great suggestion.
Ok Frank, let's do it. Here is the simplest VPython program I can think of to model the motion of a falling object.
Running this program gives:
This is essentially the same value that Khan obtained. It is a numerical calculation, so the value depends on the size of the time step. This is a vector answer based on a vector calculation. I could have run this same thing just as a scalar calculation. However, let me make one small change to the program.
The only thing I changed was the initial velocity. Instead of the vector (0,0,0), I just put 0. Here is the output.
Nope. It didn't go in. It just impacted on the surface. See. Python cares about the difference between a scalar and a vector. These kinds of errors come up ALL the time when students are creating the first Vpython calculations.
What Do I Think About Khan Academy?
I might as well add this in here. Really, I don't plan on talking about Khan Academy for a while. Too many other cool things to talk about. I think there are really two parts to learning physics. There is the part that deals with building models and there is the part that deals with looking at the models that have already been developed. I don't see Khan Academy working on the building models part. This stuff is super hard to do even in a real face-to-face classroom.
So it seems Khan Academy focuses on showing students the current models. You know, things like the kinematic equations and Newton's 2nd Law stuff. Unless you want to completely start from scratch in your model building, you have to find out about these ideas somehow. Then Khan Academy is a textbook - except online and as a video. Oh sure - there are other parts to it too, but I am generalizing. But as a textbook, it is very important that it really gets the concepts correct.
Why Do Students Love Khan Academy?
Trust me, they do. If there is one thing I learned from my video critique it's that youtubers LOVE the Khan. They love Khan Academy and will defend it in any way necessary. Why?
I suspect the problem has to do with grades. The common model (and I don't blame students for this, they have been trained to do this) is to first and foremost get the grade. They are very answer focused. That is their prize. If you have any teaching experience in physics, you know how much students love putting numbers in at the end of the problem. For them, the whole point of the problem is to get a number at the end. For them, the Khan Academy videos are the shortest path to getting this answer.
Here, I made a cartoon to show my point.
Khan is trying to help these students get more physically fit so he paces them with a golf cart. The students think the point is to get to the destination, so they just grab on the back and go for a ride.
Let's Be Friends
I know I can be a real jerk sometimes, I really do. Honestly, I am trying to help. I am not jealous of Khan at all. I think that he has put a metric TON of work into his videos and the amount he has produced is epic. EPIC. Khan Academy has a tremendous following and the videos (even with the errors) can make a huge impact. No one can deny the contributions Sal has made. And I think that is why other educators get a little annoyed. With all the effort he has put in, Khan Academy could be so much more. It is almost to the point where it would be a nice second textbook for students to reference (and not just when they get stuck on their homework).
I will be happy to help with the videos in Khan Academy. Really, I would. I know I am not perfect either, but I am willing to help. Many, many physics and math educators would be more than happy to help. So let's make it happen. Sal, give me a call. Or better yet - meet at the next American Association of Physics Teachers meeting in New Orleans. I will obviously be there since it is so close. You would not believe how many educators would be willing to meet with you to make physics education better. Go Team Physics.
This post was WAY longer than I intended.
Other Resources
If you want to talk about students learning from videos, you can't forget about this awesome video from Derek Muller (of Veritasium and @veritasium on twitter). For his Ph.D. research, Derek looked at how videos impact student learning. You have to watch this short video (which is learning about how hard it is to learn from a video by watching a video).
For other comments on Khan Academy, your best bet is to look at the work of Frank Noschese (2011 Presidential Awardee for Excellence in Science and Math Teaching). Frank has a whole bunch of posts about Khan Academy. This is required reading.
Let me just say one more thing about videos. What if the students made the videos instead of an instructor? In this case, the video can be both a form of assessment and a learning tool (students can learn a lot by making a video). I first heard about student screencasts from Andy Rundquist (@arundquist). He has some nice posts about students making videos at SuperFly Physics.
I guess you can't talk about student screencasts and videos without talking about Standards Based Grading, can you? I don't want to go into to this right now - so just go read all the stuff Shawn Cornally wrote at Think Thank Thunk. Really, that is the best place to start.
