Via Tom, a site giving problem-solving advice for physics. While the general advice is good, and the friendly, Don't-Panic tone is great, I do have a problem with one of their steps, Step 7: Consider Your Formulas:
Some professors will require that you memorize relevant formulas, while others will give you a "cheat sheet." Either way, you have what you need. Memorization might sound horrible, but most physics subjects don't have that many equations to memorize. I remember taking an advanced electromagnetism course where I had to memorize about 20 different formulas. At first it seemed terrible, and I kept remembering them wrong. However, the more you use the formulas, and the more you understand what they mean and - if you care enough to check - where they came from, the easier it gets to remember them.
Organize your formulas in front of you. If you have a cheat sheet, align it next to your variables. What formula can you fill up, leaving the least amount of missing variables? Which formula can help you solve the question?
See it? Use it.
It's not that this step is necessarily ineffective-- for certain classes of physics problems, it's actually a very sensible and efficient way to attack the problem. In fact, I used to give students a problem-solving spiel that was just a variant on this.
Over time, though, I've started to think that while this advice is good early on, it sets students (and thus faculty) up for a major problem later on. And by later on, I don't mean "in upper-level physics classes," I mean "a few weeks later in the same class."
The "find the formula with all the relevant variables" method works best for kinematics problems, which is the first topic covered in most introductory mechanics classes, and thus the first physics topic most college students see. In kinematics problems involving one object, there are six numbers you can possibly know or care about (initial and final positions, initial and final velocity, acceleration, and time), and you can effectively attack almost any problem by making a table of these six values, and determining which you know and which you care about. This will almost always lead you to the right formula to use to solve the problem in the fastest way possible.
The problem with this approach is that it sets students up to believe that physics is like a mad lib. You've got a bunch of possible formulas, and a bunch of variables that slot into those formulas, and it's just a question of fitting the right things into the right places.
This approach fails spectacularly about two weeks later, when you start doing problems about forces, for the simple reason that you no longer have a wide selection of possible formulas that you just select and plug numbers into. While the six-number table worked great for kinematics, I would constantly get students asking "Which formula do I use?" two weeks later. It's a simple question to answer, because there's only one: F=ma. That answer doesn't really help students accustomed to the pick-a-formula method, though.
While it's still true that you can pick things that feed into F=ma based on what information you're given-- if it has a spring, then you use Hooke's law, etc.-- but ultimately, the only way to solve force problems is through understanding what's going on: What forces act? In what directions? What do you know about the motion? You can eventually get the problem down to an expression that you can plug numbers into, but it requires a bunch of thought before you get there, and you have to know what you're doing.
So while the pick-a-formula method works for some things, it's kind of a dangerous bit of advice. There's really no substitute for understanding the physics of the situation, but the pick-a-formula method can make it seem like there is, and that sets students up for trouble later on.
Of course, this leaves professors in a bit of a bind, because the pick-a-formula method is effective for the early part, and even if you don't tell students to do it that way, they'll discover it for themselves. You can kind of get around it by phrasing the problems in ways that force students to do something else-- the Matter and Interactions curriculum we use now does some of this, by not giving the kinematic formulas that most pick-a-formula methods rely on, but insisting that students derive the results from simpler rules. This runs the risk of convincing students, particularly those who have had a good high school class where they saw the relevant formulas, that you're just being a dick for no real reason, though.
Anyway, this is a bit of a nit-pick regarding what is generally a solid page of advice for students. Particularly the part about not panicking. If you're taking introductory physics, by all means, read this page carefully and think about what it's telling you. But don't rely too heavily on Step 7, because while it works some of the time, it will let you down in the end.
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I suspect the "pick the right formula" line is aimed at students who have an adversarial relationship with algebra. The method still works if you can memorize enough formulas and understand what they are for, but doing physics that way becomes increasingly difficult as the term goes on, and by the time you get to E&M it is certainly not the most efficient even for the most vehement math-hater.
I'm not sure how to fix the problem of physics students who hate math, but those who can need to overcome their math phobia. Algebra is your friend: it lets you solve not just one problem at a time, but entire categories of problems. I suppose this is an argument for making sure that Physics 101 students have had some calculus beforehand: students who cannot overcome math phobia will wash out of calculus before ever getting to physics.
I work explicitly against the "pick the formula" attitude by giving irrelevant information in the question, and irrelevant formulas in the cheat sheet. The purpose of the questions is not getting to the answer, it is training yourself to think about the physics, getting to the right answer by the wrong method accomplishes nothing.
Irrelevant information is one good strategy. It also helps make the point that algebra is important-- there are a lot of intro type problems where the final answer doesn't depend on the mass, say, but students will plug it in and carry around lots of extra digits for the whole problem because they don't realize that it will divide out in the end.
Another strategy is to provide minimal information, which is what Dan Meyer advocates in this TED talk about math instruction. Which sounds like a better overall method, but is damnably difficult to implement effectively.
I agree with the original post and with Moshe. "What formula can you fill up, leaving the least amount of missing variables?" is bad advice.
Unfortunately, given the way intro physics classes are often taught, it's often very effective. At my institution, the standard in the intro classes is to assign problems in the style of: "v=#, l=#, a=#, what is t?", where the variable names exactly match the variable names in the formulae collection at the back of the chapter. I don't see how this promotes (or tests) any sort of comprehension.
I try to write problems entirely in words to stymie the "match the variable names to the formula" approach to problem solving, and I'll also occasionally include irrelevant information. The students seem to strongly dislike the latter, however.
Does that actually make the point you're trying to get across?
Having a quantity cancel out at the end of a problem is only obvious if you're already using algebra to solve it. When you solve using numbers, all answers are equally obscure. If some of the numbers cancel other numbers at the end they're unlikely to notice, and if they do notice they're unlikely to understand that they've chosen to make their task more difficult.
Students will naturally use the easiest strategy that has worked for them in the past. I've had to keep telling my General Physics 3 students not to memorize special case equations, but learn the concepts and how to apply them. My quizzes so far have been well designed because nobody is succeeding by memorizing. I only ever give the fundamental equations and I'm continually surprised by the things that students will memorize.
I am tempted to argue that weeding out the ones who think that making them derive results is dickish is a very good thing. Maybe I can convince the department to switch to M&I?
Why do people have to memorize formulas? Most people wont use that same formula their whole life. What are the most common formulas?
Hey Chad,
Thank you for this critique, I appreciate the time you took to lay out your thoughts. I read through it, and I have to admit, I see your point and accept it.
I think I failed in delivering my main idea, though.
I don't mean that people should memorize and then just pick the "easiest" equation to solve, what I mean is that if you have a set of equations, you can get a sense of what strategy to pick by examining the relationships between variables that the formulas give you.
For instance, I recently faced a freshman-level E/M question in my tutoring about the movement of an electron inside an atom (simplified, of course). The question specifically said to refer to the movement of the electron as if it is a thin circular wire with some current I. It then asked what magnetic field would be resulting from such system.
After setting up the initial steps, I referred the student to the list of equations, where we looked for two things: First, what is the relationship of the charge (Q) when it is moving at specific speed to the current (I), then which one describes a relationship to the magnetic field (B).
We found the relations and then examined what we were missing, and devised the strategy to solve the question successfully.
So, my point is not necessarily that a student should memorize the equations and then spit them out; it's more about realizing that mathematical equations give us information about the relationship between various variables. This helps with devising the strategy to solve the problems.
I completely see how this was *not* clear in this segment. I will have to think how to deliver it better, probably rephrase that segment in general.
Thanks a lot for your thoughts about this, and for all the comments. You gave me a lot to think about.
Moriel Schottlender
aka mooeypoo
SmarterThanThat.com