…assuming the teacher and student are imaginative enough and care enough to apply the needed geometry to the projection.

Since I’m familiar with the form of classical freshman notation motion functions, I can see that the given equation defines an object under constant acceleration of gravity (g/2 being simplified to 4.9 and units implied to mean m/s), but I feel for those students not enrolled in the class that talks about how to derive this equation from scratch and understand that the given equation is essential to getting a reasonable, uniform answer related to the subject matter being discussed in the preceding chapter.

If all givens on the problem are excluded, then students would have to reconcile the fact that the balloon could have vertical or horizontal components to velocity, there might be a wind in any direction, and there might be a component of wind resistance on the sandbag as it falls perhaps giving rise to an effect of terminal velocity.

But since the problem says “The position of the sandbag is determined by s(t)” many of the above issues can be assumed to be either zero or negligible (a concept which the learned appropriate application of is in a Science class than it’s purer cousin Math).

The language is quite succinct. “The position” speaks more absolutely the concepts being implied than “the height” which could leave room for a vector of movement in the plane parallel to the ground. Suddenly I’m aware that no where in this question does it say “Assume the ground is level and flat.” Where it could be further complicated by hills, valleys, or a constant angle.

The problem also does not mention the boundaries that this equation is accurate for, t from 0 through x where s(x)=0. A clever enough student in the right classroom might be able to argue sufficiently that a correct function for the speed or position of the shadow related to time would need to be properly bound to an interval of t or else it would be a constant value of zero sometime shortly after the impact, and thus necessitate at least partial credit being taken from all other average students who answered much like the book’s key presents.

But I suspect the answer listed in the teacher’s manual implies the use of a geometric translation of sandbag height to shadow distance from impact and a derivative of that to acquire an expression of linear velocity related to time. And in which case, I’d argue much like NASA does for loss of vehicle events that a failure of imagination is acceptable for humans unexperienced at a given task.

I love this! I talked about this (less eloquently) in my post on switching to something like SBG this semester and getting away from the textbook. I framed the course as linear and quadratic topics. I framed the quadratic topics by throwing chalk over and over on the first day of this part of the course, first to get them to draw the path, then to talk about height versus time (for chalk thrown straight up).

Today when we started talking about factoring, and a student asked when we’d use it, I was able to remind her of the chalk, and make up a problem that was sort of like the chalk, and factorable. (-16x^x+32x+48=0. I think the 32x term represents an unrealistic initial speed, and why would I start 48 feet high?)

Some people have argued that this is not pseudocontext because it is interesting. A couple of points on that. Someone (I think Dan) at one point said that we shouldn’t judge a problem’s merit on whether or not someone finds it interesting, because you can always find someone who is apathetic to the math or the subject of hot air balloons or whatever. If I really think about it, I could find this an interesting exercise, but again I think that it is purely from the fact that it is a Calculus problem and involves some higher-order thinking. I also found the problem of My Favorite Orange at least entertaining, even if absurd.

I also like Jason’s point, and this was really why I chose the problem in the first place: this is way down on the list of things to ask about when in a hot air balloon. Lack of relevancy of course does not make it pseudocontext, just a dumb problem. But I still think that there is something about creating some sort of context, no matter how ridiculous it is, and then asking a question that really doesn’t have much applicability to the construct you just created.

I’m OK conceding that this might not be pseudocontext in the way it has been previously shown so far. I’ll admit that this is not as glaring of an example as creating an arbitrary algebra problem out of the number of jingles on a dress. However, I still think that there is something inherently wrong about this problem. The only difference between this problem and the one with the guitar is that the equation for this is derived from the position function for an object in free fall, which is, I guess, somewhat contextual, while the system of equations from the guitar example is completely arbitrary. However, no mention of free fall is made in the problem, and the student doesn’t have to use any knowledge about an object in free fall. I think that is why this problem was such a glaring example of what I thought to be pseudocontext. Look at the original statement from Boaler: “Students come to know this about math class. They know that they are entering a realm in which common sense and real-world knowledge are not needed.” It is pretty easy to do this problem without any knowledge of the real-world and the way that gravity works (oh, and, let’s totally ignore air resistance, because that is not real world, right?). It still lays out a set of parameters and allows you to plug and chug your way to the finish line. You could replace that position function with any random equation and get to some answer and you wouldn’t know that the answer didn’t make sense for how things actually behave on Earth. Again, maybe not pseudocontext, but a different manifestation of a poorly written problem. But its all a fine line, methinks.

I actually think that one of the problems with most textbooks is the lack of any meaningful frame around the big ideas [I think IMP is a notable exception, but I have only taught 2 of the units].

Chapter 7 is not a meaningful frame. The Line Factory–while a little on the cutesy side–is a decent frame. It takes the big idea and unites all of the little concepts/skills that are a part of it under one narrative. I like how a good frame makes the connections between various concepts easier to identify.

What I didn’t like about the Line Factory was that the chapter starts out using the frame, goes off and does some other unrelated stuff for a while, and then comes back to the frame assuming that everyone remembers what it is. The second time through the unit, I wound up rewriting everything that wasn’t Line Factory into LF-material. It made for a more cohesive unit.

“My Favorite Orange” seems more acceptable because it is playful in its unwarranted application.

“Sandbag Shadow” tries so hard to be a true application that it becomes ludicrous. This isn’t even counting the “hint”.

Put another way, as I’m coming to understand pseudocontext, if there isn’t a path between your context and the math, that’s pseudocontext. The path here is rough and gravely. It isn’t a natural path. If you show your students that image and ask for questions, you’ll find ten times as many students wondering about the speed of the sandbag itself, or the force with which it’s hitting the ground, as the speed of the sandbag’s shadow. But there is a path there.

Video would do a lot to inoculate the pseudocontext. For one thing, how are you going to get students wondering about speed if you don’t show something moving? I’m thinking something like this frame. But, let’s be real, if you’re contracting a hot air balloon and setting up a camera and speed gun at the exact moment the sun is 30° in the sky, you probably have the resources to film a better related rates problem than this.

And Jason is exactly right that the givens ought to be stripped out of the problem, for a lot of different reasons, not the least of which is that you’re doing the student’s work for her when you give her that information.

It wouldn’t have to be a balloon and sandbag, but I’m not sure how else to set up this problem. If it were an object being dropped from a wall, the wall would block the shadow. What other options are there for setting up this problem?