Let’s discuss «The Third Law of Instruction» as set forth by Heller&Heller in reference 1. The «law» says:
Make it easier for students to do
what you want them to do,
and more difficult to do
what you don’t want.
Here’s the executive summary of my analysis:
In my opinion, I consider these three things to be:
These points are discussed in more detail in section 2.
|As much as possible, focus attention on important problems.||Avoid trivial problems.|
|If necessary, make the problem easier by dropping hints.||Never take a trivial problem and make it harder by adding fences.|
This is central to my idea of what teaching is. It affects how the students think about me. I do not want to be the enemy, making their life harder, erecting fences to compel them to do stuff they don’t want to do. Instead, they are paying me to be their helper, empowering them to do stuff they want to do, and do it more easily.
I tell students that I cannot possibly teach them more than a tiny percentage of what they need to know. The best I can do is instill high standards and a love of learning, and get them to a place where they can teach themselves all the other stuff they need to know. Those are the top things on my list of priorities. Also near the top of the list is reasoning in general. Note that reasoning skills are portable from one application-domain to another, as discussed in reference 2. Math skills and physics facts are on the list also, but they do not rank nearly as high as reasoning.
On pages 53–54, H&H present a worked example. Their answer is wrong by a couple orders of magnitude, as discussed in section 3. However, even if they had gotten the right answer (in physics terms), I would still object (on pedagogical grounds) to the way it was done, as we now discuss.
They intentionally and emphatically limit the student to using a small number of pre-approved equations.
|They claim this «solves» the pattern-matching problem.||I say their cure is worse than the disease.|
|They say they want reasoning rather than equation-hunting.||I say that by providing pre-hunted equations, they get rid of equation-hunting in a way that also gets rid of reasoning.|
As previously mentioned, it is crucial to distinguish between what H&H say about their «third law» and what they actually do in practice.
Imagine an aquatic training center where they keep each student inside a giant plastic sphere. Sure, this protects them from drowning, but it also prevents them from swimming. The graduates of this program think they can swim. Bad things happen when they actually try it in the real world.
Here’s a more realistic metaphor: When teaching a kid how to ride a bike, training wheels are not considered good practice. Using them is a common mistake, but that doesn’t make it any less of a mistake. You cannot learn to ride a bike with training wheels. It’s not really a bicycle at all, but rather four-thirds of a tricycle. Riding a bicycle requires leaning into turns. The training wheels supposedly prevent you from falling over, but really they just prevent you from riding properly. (In reality, they don’t entirely prevent you from falling over, but that’s a separate issue.) There is a step-by-step building-block approach to learning to ride that ensures the kid is safe at all times and in control at all times.
By providing pre-hunted equations, H&H protect the students from making any wrong decisions, in a way that prevents them from making any decisions at all. Their cure is worse than the disease. I say decision-making is part of the game. Indeed, decision-making is most of the game. By preventing the students from making decisions, you forfeit the game before it has even begun.
Let’s revisit in more detail the points made in section 1:
Make it easier for students to do what you want
them to do and more difficult to do what you don’t want.
That’s OK as far as it goes. It’s not terribly profound. It is oversimplified, but a reasonable person could take it, interpret it in a reasonable way, and derive some value from it.
On the other hand, alas, this «law» can be interpreted in less-than-reasonable ways, as we now discuss.
The metaphor is clear: The easy routes through the passes explicitly represent “pattern-matching” and “plug-and-chug" (and by extension, equation-hunting and other dirty tricks). The mountaintop explicitly represents “learning physics through problem solving”.
To their credit, H&H depict and describe provding ladders to assist the students. That’s not the problem.
The problem is related to the fences. The devil is in the details. Fences can be used wisely or unwisely. The problem is that when H&H apply their own law, they place put too much emphasis on fences, and on the wrong kind of fences.
It is necessary to think clearly about the goals. We need to distinguish
If the students solve the quiz problems using equation-hunting, there is nothing wrong with the students. I say there is something wrong with the quiz. Equation-hunting has no place in the real world, not because it is forbidden, but because it simply does not work in practice. Equation-hunting is characteristic of toy problems, and symptomatic of a mismatch between the grubby day-to-day goals and the lofty long-term goals.
As discussed in reference 2, I tell students:
We do not choose problems because they are hard, or because they are easy. We choose problems that are important.
If you can find an easy solution to an important problem, that’s the biggest win of all. That’s what we call low-hanging fruit. I love low-hanging fruit. I go for the watermelons, because they are huge and delicious and very, very low-hanging.
In this class, we do not do hard problems. We will sometimes do problems that would have been hard if you didn’t know the tricks. So let’s get started. Let me show you a few good tricks.
Of course I have to explain the difference between good tricks and dirty tricks; see reference 2.
To use the H&H metaphor: Suppose there is a small prize located in the lowlands on the far side of the mountain. To get there, any sane person would go around the mountain, through the passes. Clambering over mountain when the goal can be achieved more easily by other means is just plain silly, and the students know it.
In my opinion, the idea of fenced-off passes is the opposite of good problem-solving technique, and the opposite of good pedagogy. No amount of fences (or ladders) will improve the situation. The good option is to change the goal that is set before the students. To repeat what was said in section 1:
Do not start with a trivial problem and make it harder by adding fences. Instead, start with an important problem and make it easier (if necessary) by dropping hints.
This is central to my idea of what teaching is. It affects how the students think about me. I do not want to be the enemy, making their life harder, erecting fences to compel them to do things they (for good reason) don’t want to do. Instead, they are paying me to be their helper, making their life easier, empowering them to solve important problems more easily.
If the prize is located on top of the mountain, there is no advantage to going around the mountain. You could go around 100 times, around and around, without getting any closer to the prize.
Let’s be clear: Do not ask students (or anybody else) to climb the mountain unless it is both feasible and advantageous.
We can also address practical operational issues, not just the perceptions. Fences don’t make it any easier to get up the mountain, but there are other things that do. We can provide the students with maps, compasses, safety helmets, kletterschuhe, etc., and maybe a Sherpa or two. We can even go on ahead with picks and shovels and axes, to lay a trail that will be easy to follow. By way of motivation, we can give students a picture of the pot of gold that awaits them.
So you see, I prefer seduction to compulsion. Students do better when they are trained, not constrained.
The best use for a fence would be to fence off a small but deadly pitfall. This protects students from making a fatal mistake, but does not completely shut down the decision-making process. This leaves N−1 paths open. This is very different from the ultra-harsh restrictions advocated by H&H e.g. on page 34 and page 48.
Even more importantly, fencing around the pitfall involves closing off a route that nobody really wanted to go down anyway. The teacher’s role is helpful, not adversarial. This is very different from the H&H approach of closing off routes that people would normally and properly go down.
However, I do not buy this theory at all. For real-world problems, it is immediately obvious that equation-hunting is not an option. Putting up a big fence or a big sign that says “no equation-hunting allowed” will not make it any more obvious than it already is.
If you are worried that students who are addicted to cheap tricks will simply give up, rather than climbing the mountain, well, that’s a problem, but fences will not help, not even a little bit. They will give up when they get to the fence ... instead of giving up an inch sooner or an inch later, when they see that equation-hunting was never going to work anyway. More practical ways of persuading students to climb the mountain are discussed in item 2 and item 4.
Training expeditions of this kind do not require fencing off the passes.
My point is that there is not a binary choice. There are shades of gray, and even colors. Mindless equation-hunting lies at one corner of the parameter space. Super-scary challenging problems lie at the opposite corner. There is a lot of space in between, and that’s where the wise strategies are to be found, far from any extreme.
As an exercise, let’s imagine that the left pass is closed by flooding, and the right pass is closed by an enormous wildfire. We need somebody to take some life-saving serum to the villagers on the far side. Can you find a path over the mountain?
We are not asking you to take the hardest path. Indeed you are encouraged to find the easiest remaining path. Only the two most-obvious most-familiar paths are blocked. You are allowed to use all resources at your disposal, including everything you know, except for the two blocked routes.
This scenario allows the student to build up skills and build up confidence that will be useful if there ever is a real emergency.
The scenario is important here. The instructor needs to construct some sort of scenario to justify the extra effort. There is an explicit acknowledgment that the route through the passes makes sense in non-emergency situations. Taking the difficult path is not normal, and certainly doesn’t get elevated to the status of a «law».
Emergency scenarios, by definition, are not normal. The normal reason for climbing a mountain is because there is something you want up there. Most of the training should be focused on the normal situation.
It is important to design the scenario around a short list of things that are disallowed, rather than a short list of things that are allowed. This is crucial for developing reasoning and judgment.
Abnormal scenarios need to be carefully calibrated. As the proverb says, a fool can ask ten questions while the wise man is answering one. By that I mean that the instructor can all-too-easily construct a scenario that leaves the student exhausted and terrified. If at all possible, the level of difficulty should be titrated on a per-student basis. As a starting point, it pays to ask the student: Today would you like a back-to-basics review lesson, or would you like something more interesting and more challenging? Similarly, during the course of the lesson, it pays to ask: How’s your workload? Would you like me to leave you alone, would you like some help, or would you like me to add some more excitement to the scenario?
If you start with a challenging lesson and they make a hash of it, you need to quickly pivot to reviewing the basics.
In all cases, be careful not to put too much emphasis on emergency scenarios. You don’t want students to get too comfortable with operating at the edge of the envelope. It is important to know where the edge of the envelope is, and to know how to operate there if necessary, but in non-emergency situations you want to stay far away from the edge.
When dealing with the highest level students, the instructor doesn’t need to block the passes, because the students will do it on their own. They bring their own fences. Neil Armstrong repeatedly crashed the lunar module simulator by taking it outside the design envelope. He wanted to explore the actual envelope, not just the design envelope. That is, he wanted to know exactly what the lander was capable of in an emergency. Guess what came of that on 20-July-1969.
In this case, no amount of fences and no amount of ladders will persuade the students to climb.
In such a case, we have to improve the reality before we have any hope of improving the perception.
You have to be careful when discussing fences in the abstract, because some fences are good and some fences are bad. In particular,
In non-metaphorical terms: Only the most oversimplified, overconstrained, ivory-tower busywork problems are susceptible to equation-hunting. Realistic problems are not susceptible, because there are too many equations, and too many ways of permuting and combining the equations.
On page 34 of the book, it says:
The only formulas that may be used are those given below.
On page 48 it says:
To compel students ... we only allow our students to use equations chosen by the instructor
That is quite an extreme position. This requires us to see the «third law» in a very harsh light. This is the sort of thing that gives Kadavergehorsamkeit a bad name.
It must be emphasized that when I argue against this extreme, I am not arguing in favor of the opposite extreme. In fact, as usual, all the extremes are wrong.
Choosing a minimal set of pre-hunted equations is unreasonable to begin with, because not all students will solve the problem the same way. It’s impossible to know in advance what equations they will need.
Even if only one method of solution were possible, the H&H scheme of relying on pre-hunted equations wouldn’t work for real-world problems. Even a rather modest real-world problem might involve a dozen equations and 20 steps. Even if some oracle told you which equations to use (so you could ignore the other thousands of equations in the world), just choosing the order in which to apply the equations leaves you with 1220 possibilities. That’s enough to make equation-hunting completely infeasible ... aside from the fact that in the real world there is no such oracle.
Also, just to rub salt into the wound: Checking the result requires more equations than were needed to derive the result. By limiting the students to the minimal number of equations, H&H prevent the students from checking the work. This is a big deal, because checking the work is fundamental to any notion of critical thinking.
Furthermore, even after you have derived the result and checked the result, you should keep thinking about it, to see how it is connected to other stuff you know. This includes looking for interesting generalizations. Over the long term, this is what makes you smart, as emphasized by reference 3 and reference 4. All this is impossible if you are given the minimal amount of information.
The larger point I am making has been part of the pedagogical literature for 2500 years that I know of, and perhaps longer:
H&H seem to think this is simple. I don’t. I think it takes skill and judgment and a certain amount of artistry to strike the right balance.
As previously mentioned, it is crucial to distinguish what H&H say from what they do. Let’s take a look at how they interpret their own «law», i.e. how they use it with their own hands. It’s a horror show.
Specifically, let’s look at the worked example on pages 53–54. This involves a pickup truck pulling a trailer. Let’s see how it plays out. How do I love thee not? Let me count the ways:
|FfT = µ FNT (2)|
Where’s That From? They say the formula applies to «kinetic friction» by which I assume they mean sliding friction ... but these tires are not sliding.
H&H distinguish two types of friction, namely «kinetic friction» and «static friction». In fact there are at least four types:
If all 8 tires were skidding, the driver would not want to accelerate from 40 to 60 mph. Really not. Indeed, it’s not even possible, as discussed below.
This is equation-hunting of the filthiest kind. Somebody hunted up an equation without regard for what it meant. To say the same thing another way, they plugged in a value that is wrong by about two orders of magnitude.
My point is, I shouldn’t have to guess. Standard good practice is to provide a legend or glossary that explains such things. See the discussion of names in reference 5 and reference 6.
According to my calculations, if we pretend the tires are sliding in accordance with the calculations of FfP and FfT given in the book, the power required to overcome sliding friction at 40 MPH would be about 395 HP, well in excess of what the truck is capable of. The system could not accelerate from 40 to 60; indeed it could not even maintain 40 under the given conditions.
In the real world, correct values for truck mass and engine power are readily available. In the real world, correct formulas for the rolling resistance are readily available, and order-of-magnitude estimates should be known “closed book” by anyone who has ever operated a car, bicycle, or tricycle.
On the other hand, perhaps the equation is trying to talk about the scalar magnitude of FTP and FPT ... but in that case it really ought to show explicit absolute-value bars.
So this is either a sign error or a third-law physics error or both. It also counts as a failure to define clearly what the symbols mean.
This would have caught the error in the previous item. This counts as a second error, insofar as the authors had two bites at the apple and missed both times.
In the limit where the mass of the trailer goes to zero, this would easily have caught the erroneous minus sign in the final result. Again there were two chances to avoid the error (during the initial algebra phase and again during the check phase).
Another weakness is the failure to mention other kinds of checks.
At this point some might argue: Those are just physics mistakes. We don’t care. This isn’t really a physics book, but rather a book about pedagogical technique. Besides, probably nobody paid attention to the example on page 53, so no harm done.
Well, I’m not buying that argument. First of all, if you write a book and nobody pays attention to what it says, you have failed in all your objectives.
Furthermore, the book by its own terms (page 6) says we are supposed to care about the physics.
Also, equation-hunting, not checking the dimensions, not checking the limiting cases, and not noticing that the answer is off by multiple orders of magnitude and a minus sign counts as more than a physics mistake; there are some rather basic reasoning mistakes in there as well. I mention this because the book by its own terms (page 6) says that in addition to physics, the goal is to teach “logical reasoning” applicable to “new situations not explicitly taught by the course”. If the authors, following their own methods, come up with nonsense results, it means the methods are no good.
What’s worse, if we assume that hundreds upon hundreds of students had the opportunity to catch these errors and didn’t, it means that the course has consistently failed to teach good reasoning skills.
Please, folks: we can do better than this.