In December I wrote a post called Pro vs. Khan which sparked many angry comments. My intent was to critique the content on the famous Salman Khan Academy site and to demonstrate what I considered to be a more effective and interactive math video.

I recently attended the NCTM math teacher’s conference, and I was pleased to see real teachers who have created some great websites that can get kids excited and motivated to learn about math. Internet math star Dan Meyer did an excellent presentation about how videos can be used to get kids thinking about math — not instructional videos, but videos of something real happening which can spark discussion. He announced a new website 101qs.com (for 101 questions) where people can upload interesting photos or videos and others can leave comments with good thought provoking questions. Another great site that is under development is from Mathalicious, and is called math52.com. What I like about both of these is that they are not just taking the same old boring thing and making tutorials — they are working on making math something that kids WANT to do. With innovative sites like these, I actually haven’t heard much about Khan Academy recently.

I’m not very optimistic about this new ‘flipped classroom’ model where students watch the lesson for homework and then do their homework as classwork in class. As someone who has been using computers since I got my Atari 800 computer for my Bar Mitzvah in 1982, I feel like I’m the sort of person who would have a good chance to learn something on a computer if I wanted to. A few years ago I was taking piano lessons and learning a bit about music theory and decided I wanted to train my ear to recognize the different types of chords and chord progressions. I researched and purchased what seemed to be an excellent software package called ‘The Earmaster.’ It had a great interface. It had questions and played the chords and tracked which ones you got correct and adjusted the questions accordingly. But after a few days, I found myself not interested in the program anymore and that was the last time I used it. And I really was motivated to do this so I can’t imagine how a kid who can’t get inspired by an actual person sweating it out ten feet away from them is going to be able to maintain concentration for a computer. One of my co-workers said she learned Spanish from the Rosetta Stone software. Maybe certain people have the discipline to learn from a computer. I don’t think, though, that everyone can.

A few months ago Khan was the keynote speaker at The Celebration Of Teaching And Learning. When someone asked about the quality of the videos, he said that the videos are unscripted and he felt that this was part of what made them popular. Even when he makes mistakes or leads students, inadvertently, down a ‘dead end,’ it is part of the process of seeing how math people think.

While I do appreciate the spontaneity and the energy that can come from improvisation, some students (perhaps most) will learn more (outcomes, right) from a lesson that is carefully planned. In my original post, I made a video to contrast with Khan’s on the same topic. Some people did not like my video and I admitted that I didn’t think mine was that great either, just better than Khan’s.

So I’ve made another attempt. In the original post, I took an elementary concept of averages and many people were critical of mine for being too slow and detailed. I think, though, that it is easy for people to think that when they already know the topic. For this one, I’m going to do a throwdown with Khan on a topic that most people have long forgotten — The Law Of Cosines. My challenge is for people to watch Khan’s thirty minute improvised explanation and derivation and then try to answer these two questions. Then watch my video and see if you can do it.

These are two questions that can be solved with The Law Of Cosines. As a pretest, take a crack at them before watching either set of videos.

Question 1: Two sides and the angle between them are known. Find the length of the third side.

Question 2: All three sides are known. Find the measure of angle C.

Answers at the end of the post.

This is a very popular topic, often worth up to 10% of the Trigonometry Regents exam, despite being something that teachers generally only spend about 4 or 5 days on.

One of the complications I had to resolve in creating my video is that different students like different types of explanation. The college way to teach something like this would be to show the ‘long’ way of doing the problem, then derive the formula, and then show how to apply the formula. You will see that Khan does the first two of these components, yet never shows how to do a practice question with the formula.

I know that some students like to see the long way and the derivation and that helps them understand the formula better. Other students like to first see how the formula works and then learn why it works later. And many students just want to get the answer right and, at least to not alienate those students, it is best to teach it that way. So what I did was add buttons that enable students to skip the derivation and get right to the formula. I also gave options to skip over review material, and also buttons that wait for the student to answer questions.

Here is Khan’s 3 part 30 minute explanation of The Law Of Cosines. Watch it and then try to answer the two questions above.

Here is my video. By skipping ahead, you can get through it in about 10 minutes. Or, you can watch the whole thing with the ‘long’ way and the derivation of the formula in about 30 minutes. Watch mine and then try to answer the two questions (you’ll need a scientific calculator — click here to access an online one)

I still don’t think that the ‘flipped’ classroom is a great way to learn math. But if its going to be done and popularized, the videos need to be high quality. I don’t think that I ‘nailed’ this lesson, but I do think it is at least pretty good. I’m interested, though, if the Khan defenders will be able to do so again when comparing these videos.

Answers:

1. 7.9 cm

2. 84.3 degrees

Sal Khan (TIME’s 100 most influential -2012)

You are going to have to yell a bit louder, because no one is hearing you.