Yet another lesson on critical thinking in math came from my calculus classes today. Actually, make that two lessons about the same idea.
Lesson 1: The tale of the tangent line. Students are given that the tangent line to a function f(x) at the point (4,3) also goes through the point (0,2). They are then asked to find the values of f(4) and f'(4).
What should happen: You ask yourself the two right questions in this situation: What does f(4) actually mean, and what does f'(4) actually mean? The answers tell you what to do. The value of f(4) is the height of the graph at x = 4, and that’s given to you because it says the line is tangent to f(x) at (4, 3) — so f(4) = 3. The value of f'(4) is the slope of the tangent line at x = 4. Well, I am given two points on this line, so its slope is easy to compute: Just rise over run, or (3-2)/(4-0) = 1/4.
What did happen: Most students computed the slope first, but they didn’t realize that this slope is f'(4). Instead, they used it to find the equation of the tangent line (y = 1/4x + 2); then load that into the limit formula for the derivative; then crank through about half a dozen lines of limit-taking to arrive at 1/4.
I don’t think it dawned on most students that when you start with a linear function and take its derivative, you find its slope — which is perfectly obviously available in the formula for the linear function to begin with. Also, most students said that the tangent line equation itself was f(x), which is wrong. We don’t know the formula for f(x), don’t need to, and are making a huge, incorrect assumption when we say that f(x) is equal to its tangent line.
Students also used the “formula for f(x)” to calculate f(4), namely by plugging in x=4 and getting 1/4(4) + 2 =3, which again could have been cherry-picked from the info in the problem statement.
Lesson 2: The moving particle. Students were working in groups on an exercise involving a particle moving in a linear fashion with a certain position function s(t). It’s one of those problems where the particle is at rest a couple of times and is moving forward and backward elsewhere. They were asked to find the times when it’s at rest, the times when it’s moving forward and backward, and finally the total distance travelled from t = 0 seconds to t = 8 seconds.
The first two questions were pretty easy for the students (although most didn’t know how to solve an inequality for the forward/backward question). But when it came to finding the distance travelled, the majority approach surprised me: Nearly everybody in the class took the derivative to get velocity, and then set up the integral of velocity from 0 to 8 and integrated. Keep in mind that this is a first-semester calculus class and we don’t hit integrals until around Thanksgiving — so this is a case of students trying to pull out information from their high school courses.
Two things about this. First of all, integrating just the straight velocity function doesn’t give the right answer — it gives the total or net change in position, which is different than the distance travelled (because the particle sometimes travels in the negative direction). Secondly, and worse IMHO, is that this approach is tremendously more complicated than it needs to be and results in doing a bunch of work (differentiating s(t)) that you ultimately undo (integrating the derivative of s(t)).
But students were out to sea about how to approach the problem without integrals. I gave a quick mini-lesson on why it’s important to visualize the problem in order to understand it; rather than particles in an accelerator, visualize a person walking up and down a hallway. How would you calculate how far they’d travelled? Pretty quickly, once the visual is correct, you realize it’s just a matter of adding up the distances between the rest points, and nothing harder or more high-tech than that. But what’s the first instinct? Throw algebra and complicated calculus at it.
What these two episodes have in common is they show the disinclination to solve problems through the simples possible means. Automatically, it seems, the approach to solving the problem is to pull out the heaviest machinery, the most complicated and powerful techniques, and compute… compute… COMPUTE… until something falls out. And by then, you’re so tired that the probability that you will check your work by trying to see the result from a different angle is pretty much zero.
It seems to me that an approach to either of these two problems that embodies critical thinking will try first to find the simplest approach to the unknown by bringing one’s understanding of the problem to bear — for example, by asking oneself first what f(4) and f'(4) really mean, and then seeing how much information is available to compute those values, and then — ONLY then — bringing in extra mathematics to compute/construct what wasn’t given to you. In this case, everything was given to you, and bringing in extra “stuff” like the equation of the tangent line and the limit definition of the derivative just amounts to a waste of time (plus wrong answers).
Actually, what these episodes may embody is the larger issue of making sure you understand the problem before you begin to solve it. Students hate this, in my experience. They are trained that the right way to solve a problem is to get it done as fast as possible and then get on to the next one. Taking time to examine a problem — to understand its terms, to visualize it, to see the real essence of what makes it a problem, to pare away the stuff in the problem that isn’t important — smacks of taking longer than you need to take. But in fact it saves time later, because it leads you to the simplest solution. (I again blame standardized testing for the “speed solution” culture that high school students now inhabit, but that’s another axe to grind.)