Prompted by a video of math teachers watching Khan Academy, another math teacher and a media theorist have put up a small prize for similar critiques of the Khan’s educational videos. The first such video appeared hours later, by physics teacher Joseph Kremer. Watching it, I realized Khan’s approach is more flawed than can be fixed by an updated video.

What Khan doesn’t realize – or if he does, it’s not clear in the video – is that there are many building blocks to what he’s doing that he just expects the viewer to have. It was the small things that tripped up the impersonated student. Yes, the Khan Academy has an elaborate prerequisite system, but since most users are teens frustrated on homework problems googling for help, who find the videos on YouTube and not on Khan’s website, there’s no guarantees about who’s watched the “required” videos. This is why the continuity in the classroom is so important.

There’s a moment where Khan is equivocating about time vs. change in time. Kremer makes jabs at this, and his feigned confusion is justified. The idea of change was a fundamental idea in physics that Khan has neglected to give its own video. I wonder if Khan is aware of the importance of the distinction between a quantity and change in the quantity, because if he was, I doubt he’d be so flippant about it. As I stewed about the poor quality of Khan’s video, I realized that I could present this idea very well – but I’d need a classroom to do it.

Before I begin, a disclaimer: I’m not a teacher. I’m just a computer scientist who thinks he can teach, just like Salman Khan.

It’s the first day of school, and the newly-christened high school juniors walk into your classroom after three periods of having the syllabus read to them, and expecting more of the same. But no. You declare that “physics is the science of motion” (or pick your own noun) and ask for a volunteer. Joe comes forward. Declare that “I want to know how tall Joe is,” and produce a tape measure. Make some excuse to get him to stand on a chair (“so I can see the tape measure better”). Measure from the floor to Joe’s head, and proudly announce his “height” to the class.

When the class protests, ask them what’s wrong. When they want you to measure from the base of the chair to Joe’s head, say that you can only measure from the floor. As much as possible, use student ideas and questions as an excuse to do the following. (“Okay, a diagram on the board? What should it look like?”) Draw a vertical line on the board with horizontal ticks on each end for the floor and Joe’s head, with another tick about a third of the way up for the base of the chair. To the left, label the entire line as the “height” you discovered. Still on the left, indicate Joe’s actual (and unknown) height as the distance from the chair’s tick to the top tick. Use the diagram to show that you should measure the distance from the floor to the seat of the chair, and do so, and label it left of the line. Subtract the chair’s height from the total height to get Joe’s actual height. While the students are working through that sequence, keep Joe on the chair, for both the novelty and the comparison between the concrete and the abstract.

Tell Joe thank you and ask for another volunteer, Jane. Produce an alarm clock and innocently ask, “how do I measure her age?” After the class thinks about it, say “Well it’s 2012 (or whatever) now,” and label on the right the top tick as such. “And we started counting in year 0,” and label the bottom tick 0 on the right. (“You don’t look 2012 years old.”) Ask Jane her age, and label that as the distance from the middle tick to the top tick. (“I don’t think this diagram is to scale.”) Get the class to tell you that the middle tick is Jane’s birth year, and see if they can figure it out.

Have Jane sit down and ask “Did we just do the same thing twice?” Encourage both yes and no answers, since the unknowns were different but the equation was the same. Then, “Remember that I can only measure height starting from zero. Does Joe have a height?” If they’re stumped, observe that Joe’s height is the change from the height of the chair from the floor, from zero, to the height of Joe’s head from the floor. Ask what would happen if you used a different chair. Conclude that “the distance from Joe’s feet to his head stays the same no matter where I start measuring. So how do I get rid of the height of the chair, which I don’t care about?” From there, have the students formalize in symbols the subtraction they did earlier in numbers. Write, probably in English, that the change in height equals the final height (of Joe on the chair) minus the initial (of the chair).

Then do the same for time. If talking about Jane’s age isn’t working, try asking “is five minutes a time or a change in time?” Prod with “Is ten o’clock a time? Ten hours since midnight, which we’ll call zero, right? Is ten-oh-five a time? What’s the change between ten o’clock and ten-oh-five? So is that five minutes a time or a change in time?” Ask students about the difference between a time and a duration. Get them to write the formula on the board, next to the one for distance, about change in time. Also point out that it doesn’t matter whether you use years or minutes to measure time.

Next have the students make the leap from change in distance and change in time to change in any quality. This is the most abstract, what a computer scientist would call *parametric polymorphism*: an understanding of change that takes many forms depending on a parameter (the physical quantity changing). (Also, now may be the time to introduce delta and subscripts.) Now go the other way. “Later on we’ll learn about a physical quantity called voltage. You don’t need to know what it is yet, but what can you tell me about change in voltage?” Notice how we’ve switched from inductive reasoning (specific to general, or concrete to abstract) to deductive reasoning (general to specific, or abstract to concrete). It’s roughly analogous to “model building” and “model deployment,” which suggests there’s room to add “model failure”. Definitely cover the degenerate case where the initial whatever is zero, pointing out that this is often what causes the confusion in the first place. Then you can cover the experimental methods. What happens if Joe slouches? What happens if Jane has a birthday next month?

You don’t have to keep the induction/deduction from the kids. In fact, do some meta-analysis and talk about how, in addition to learning a basic piece of physics, they’ve also gotten a taste of how physics is done. See if they can identify the salient processes. And then, since you won’t be able to sustain the theatrics all year anyway, read them the syllabus.

If I can do my own meta-analysis, it’s the little things that trip up students. As soon as a teacher glosses over something basic that a student doesn’t get, implying they don’t understand the obvious, the “I can’t do math/science” sentiments are born. It doesn’t matter if it’s fairly simple knowledge, although as we saw, even a little delta carries a lot of information! Change is one of the fundamental building blocks of physics, and it appears everywhere. The good news is that teaching the basics really well ought to start making headway on the “real” course material without the students even knowing it. Spend a week on the sort of stuff that Khan breezes past. (Day 2: mph vs miles/hour leading in to fractions of fractions and ratios. Day 3: Units, quantities, and conversions. SI. Day 4: How many cubic centimeters in a cubic meter? and uncertainty: if I’m off by +/- 1cm per side, what is the range of possible volumes?)

But these are not the sort of things that can be made into videos. First, the lesson plan above included room for flexibility and asked the class to do most of the intellectual work. This is because a lesson is not something that can be bottled and mass-produced; the lesson occurs in the mind of the student. By this credo, the Khan Academy is fatally flawed. But even more broadly, the internet’s must-have-now culture ensures that videos about necessary building blocks will languish unwatched while students dive into the material relevant for their problem set. The internet does not promote the patient buildup of ideas only indirectly related to the problem at hand, at least when it’s a homework problem. It’s not just this lesson but this basic-first methodology that can only work in a classroom.

Even more so than usual, feel free to comment…

Also: anyone teaching students themselves in a traditional classroom may use this lesson plan. If you’re a publishing company or the Khan Academy, email me for approval (see the About page).

Posted by Luis M. Aguilar on July 14, 2012 at 2:24 pm

I have been teaching for more than 25 years, and totally agree with you. I found your lesson description overall interesting, and will give it a try.

Posted by Max Goldstein on July 15, 2012 at 5:39 pm

Really? Why thank you! Please let me know how it goes and any changes you make.

Posted by Polly Brizzi on July 24, 2012 at 11:17 am

Point well taken! I am in the same demographic as Mr. Aguilar and have the challenge of mentoring a first year physics teacher this fall. Your lesson helped me remember the most difficult lesson that I learned over my “rookie” years in the early 80s. I had an adequate mastery of teaching methods, but was unaware of the need to adapt my instruction to meet the needs of both “microwave” and “crock pot” learners. As I try to pass along what I know to my younger colleague this year, your lesson provides a great example. The “microwave” learner (Khan) intuitively sees the distinction and sees no need to make it a significant part of the discussion. The “crock pot” learner (The” I Just Don’t Get Physics” Kid) needs to make the connection before he/she can proceed on to the multi-step analytical skills needed to master problem solving. Thanks for the idea!

Posted by Max Goldstein on July 24, 2012 at 10:10 pm

You’re very welcome! One of the key problems with Khan is that it can’t address different types of learners. Some kids can get final minus initial immediately, others can get it when prompted to think about the different between time and duration, and still others need it done to death like I describe. With a concept so fundamental, even the fastest learners benefit from having a firmer foundation. It also illustrates what’s really going on: rather than a formula to memorize, it’s an inescapable, logical law of the Way Things Work ™.