www.findthederivativewithstepsfree.com<\/a> (not, so far as I know, a real website, but a thirty-second Google search will give you a plethora of possibilities).<\/p>\nTranscending the Machine<\/strong><\/p>\nThe good news is that computers are good at exactly one type of problem:\u00a0 problems that have algorithmic solutions.\u00a0 If you can describe<\/em> the exact sequence of steps needed to solve a problem, then a computer can implement those steps faster, more accurately, and more cheaply than any human being.\u00a0 The real moral of the story about John Henry is don’t compete with the machine in the machine’s areas of strength.\u00a0 <\/em>Instead, find the things the machine is bad<\/em> at.\u00a0 In this case, the easiest way to neutralize these problem solving sites is to make every problem a word problem.<\/p>\nOf course, \u201cstudents can’t do word problems.\u201d\u00a0 This is a meaningless objection:\u00a0 at the start of calculus, students can’t integrate, but we still ask them integration questions on the final exam! Our job is to teach these students how to do these things.\u00a0 Here’s where the flipped classroom becomes a key part of the solution.\u00a0 Don’t<\/em> spend class time lecturing:\u00a0 students can view lectures on their own time.\u00a0 Instead, class time should be spent working problems, especially those that can’t be solved by following a sequence of steps.<\/p>\nIt’s helpful in the discussion that follows to think of problems as falling into one of two categories:<\/p>\n
\n- Routine problems, where the mathematical question and the relevant information are explicitly given: \u201cFind the derivative of y<\/em> = tan(3x<\/em>).\u201d<\/li>\n
- Non-routine problems, where this information is not given explicitly. Roughly speaking, every word problem is non-routine, and such problems form the bulk of the questions in \u201creform-oriented\u201d textbooks.<\/li>\n<\/ul>\n
<\/p>\n
Flipping Your Class, Social Distancing Edition<\/strong><\/p>\nHere’s one possible structure for such a class (where \u201cclass\u201d means any time you’re working with students in realtime).\u00a0 All of the following takes place before class:<\/p>\n
\n- Students watch assigned lectures on the topic.<\/li>\n
- Students complete routine homework problems, using some online homework management (OHM) system. If you’re using a commercial text, there is an OHM associated with your text.\u00a0 If\u00a0 you’re using your own, there are free products (MyOpenMath is my go-to) that can be used.<\/li>\n
- Students are also assigned a set of non-routine problems to consider. These don’t have to be separate from the OHM:\u00a0 again, almost every word problem should be considered non-routine.<\/li>\n<\/ul>\n
How should you run class itself?\u00a0 Class time is the most valuable resource available to students; using it efficiently and effectively can be challenging.\u00a0 Here’s a few things that may help.<\/p>\n
At the start of class (online or in person), take down a list of student questions.\u00a0 One risk is that the more outspoken students tend to dominate the discussion; taking down a list of all questions at the start of class is a way to make sure that every student has a chance of getting their question answered, and to ensure that a sufficient variety of problems are presented.<\/p>\n
Establish from the start that the routine problems have lowest priority:\u00a0 these are problems that should be solvable by students who followed the assigned lecture.\u00a0 It is vitally important that you keep to this rule:\u00a0 The biggest challenge to running a flipped classroom is students who don’t watch the lectures beforehand.\u00a0 Depending on how you’ve set things up and the system you’re using, it might even be possible to determine whether a student has watched the assigned lecture (though trying to do this realtime requires a bit of practice); another option, which I use, is to assign simple 1-point problems that students answer after they’ve watched the lecture.\u00a0 Remember:\u00a0 Class time is the single most valuable resource available to the students; it should not be spent on things that can be done out of class time.<\/em><\/p>\nOne way to efficiently use class time is to focus on the setup.\u00a0 \u00a0\u00a0For example, let’s consider the following problem, which probably appears in every calculus text ever written:<\/p>\n
A 25-foot long ladder rests against a wall.\u00a0 The base of the ladder begins sliding away from the wall at 2 ft\/second, while the top of the ladder maintains contact with the wall.\u00a0 How rapidly is the top of the ladder falling when the base is 10 feet away from the wall?<\/em><\/p>\nThe \u201csage on the stage\u201d would identify the relevant parameters and write down the mathematical problem to be solved.\u00a0 The \u201cguide on the side\u201d would lead students to the mathematical problem.\u00a0\u00a0 For this, it’s important to ask leading questions and not give outright answers.\u00a0 For example:<\/p>\n
\n- What’s going to answer the question \u201cHow rapidly?\u201d (Students should identify that this is an instantaneous rate of change, so it’s a derivative)<\/li>\n
- What other things are changing? (Students should recognize that the distance of the base of the ladder from the wall is also changing, but the length of the ladder is not)<\/li>\n
- Is there a relationship we can write between the quantities?<\/li>\n
- Which derivative do we want? (Students should identify that they want $\\frac{dy}{dt}$; it’s also worth making them explain why $\\frac{dy}{dx}$ is not relevant).<\/li>\n<\/ul>\n
In the end, we have the mathematical problem, \u201cFind $\\frac{dy}{dt}$ when $x^2 + y^2 = 25$ and $\\frac{dx}{dt} = 2.$\u201d\u00a0\u00a0\u00a0\u00a0 At this point, it becomes a routine problem—and if you’ve established that minimal class time will be spent on routine problems, you can leave the problem at this point, perhaps with a directive of \u201cFinish the problem after class.\u201d<\/p>\n
It’s worth noting that, at this point, the problem can be handed off to an online calculator, which can then solve the problem.\u00a0 You might even go so far as to point students to the online calculator, lest they develop a mistaken belief that you’re unaware of the existence of such things.\u00a0 This epitomizes the idea that humans should do what humans are good at, namely extracting the mathematical problem to be solved; while machines should do what machines are good at, namely applying an algorithm.<\/p>\n
As the preceding example suggests, it’s possible to teach a flipped class with very little change in how you’re already teaching.\u00a0 The main difference is establishing the expectation that students watch lectures before class.<\/p>\n
Let’s see how we might take a larger step, using a standard topic:\u00a0 finding the extreme values of a function.\u00a0 A traditional approach might be to have students find derivatives, then critical values, then apply some test to decide whether a critical value corresponds to a maximum or minimum.<\/p>\n
In a flipped classroom, students wouldn’t be given this algorithm.\u00a0 Instead, they’d create their own approach, typically through some guided exploration of a question.\u00a0 Coming up with good questions is challenging; fortunately, thirty years of reform calculus have provided us with an abundance of material, and many of these questions have been incorporated into every standard calculus text, so you needn’t write your own.<\/p>\n
For example, I like to give students the following question:<\/p>\n
An accelerograph records the acceleration of a train (assumed to be moving in a straight line); some of the data values are shown below.\u00a0\u00a0 Assuming the velocity of the train at t = 0 was 0 m\/s, estimate when the train was moving the fastest; defend your conclusion.<\/em><\/p>\n\n\n\n| t (seconds)<\/td>\n | 0<\/td>\n | 1<\/td>\n | 2<\/td>\n | 3<\/td>\n | 4<\/td>\n<\/tr>\n |
\na(t) (m\/s2<\/sup>)<\/td>\n| 3<\/td>\n | 2<\/td>\n | 1<\/td>\n | -1<\/td>\n | -2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u00a0<\/em><\/p>\nA sequence of leading questions can guide students to creating their own approach:<\/p>\n \n- What does the acceleration have to do with the velocity? (The student should identify that it’s the rate of change of the velocity)<\/li>\n
- So a<\/em>(0) = 3 and a<\/em>(4) = -2. What does that tell you?\u00a0 (The student should identify that the velocity is increasing at t= 0, and decreasing at t = 4)<\/li>\n
- So when is the velocity increasing, and when is it decreasing? (The student should identify the velocity’s increasing at t = 0, 1, 2, and decreasing at t = 3, 4.\u00a0 I usually find I have to ask \u201cIs the velocity increasing or decreasing at t<\/em> = 1?\u00a0 t<\/em> = 2?\u00a0 )<\/li>\n
- So where is the velocity going to be the greatest? (The student will probably<\/em> say t<\/em> = 2, at which point remind them that they just told you velocity is increasing at t = 2)<\/li>\n<\/ul>\n
and so on, leading to an answer like:\u00a0 The train’s velocity appears to be increasing until at least t = 2, and is decreasing from t = 3, so there’s a maximum velocity between t = 2 and t = 3.<\/em><\/p>\nWhat’s worth noticing here is that none<\/em> of these questions can be answered by appeal to a formula or an algorithm.\u00a0 Consequently, any attempt to use an online calculator on this type of question will result in, at best, a nonsense answer.\u00a0\u00a0 The closest thing to an \u201calgorithm\u201d is recognizing that the change from increasing to decreasing is where the local maximum value will occur, but even then, since that change occurs \u201coffscreen\u201d, students must consider how they know that the change has occurred.<\/p>\n <\/p>\n A Return to Normalcy<\/strong><\/p>\nSuppose, against all predictions and the entire trend of human history, we go back to how things were at the beginning of 2020:\u00a0 traditional in-person classes, no social distancing, exams where we could control the resources used by students.<\/p>\n None<\/em> of the preceding needs to change.\u00a0 In fact, all<\/em> of the preceding alterations in our pedagogy and our assessment are worth doing regardless of how we will give exams.\u00a0 The hard truth is that sooner or later, our students will leave the classroom.\u00a0 If what they’ve learned from our classes can be done by a free internet application, then their education is worth a free internet application.<\/p>\nWe owe it to our students to give them something more.<\/p>\n \n | |