Advanced Mathematical Concepts: Precalculus with Applications. Glencoe. 2006.
Pseudocontext
It asks students to ignore reality in order to solve it. The wind would probably be moving the balloon as the balloonist tried to take these measurements, and it’s unlikely that the two angles could be measured from exactly the same vantage point. The problem also assumes that Groveburg is a flat city, or at least that the elevation of the soccer fields is the same as that of the football field.
And, while not pseudocontext, we have another situation where the author asks the student to solve for inconsequential measurements and ignore the consequential one:
The questions aren’t what you want to know. You want to know how high up the balloon is! It’s not like you can’t figure it out from the info given either. It can be done with the Law of Sines, which is the focus of the lesson.
And then we’re back to the pseudocontext:
If you do calculate the balloon’s height, you find that the balloon is about 1.24 miles above Groveburg, which is also unlikely since an average hot air balloon ride only goes up to 2000 feet.
Transcription:
A hot air balloon is flying above Groveburg. To the left side of the balloon, the balloonist measures the angle of depression to the Groveburg soccer fields to be 20° 15′. To the right side of the balloon, the balloonist measures the angle of depression to the high school football field to be 62° 30′. The distance between the two athletic complexes is 4 miles.
- Find the distance from the balloon to the soccer fields.
- What is the distance from the balloon to the football field?
Assignment:
- Scan an example of pseudocontext.
- Email it to dan@mrmeyer.com
- List the textbook title, edition, and publisher.
- Give me your interpretation of the term “pseudocontext.”
- Let me know if you’d like credit (name, blog or twitter) or if you’d prefer anonymity.
10 Comments
Curmudgeon
December 5, 2010 - 8:49 am -I agree that the distance from the balloon directly to the field is useless and the height being 1.24 miles is equally silly.
The measurements could be made easily, though, since the balloons are never traveling very fast, usually less than 5mph. Secondly, the direction of travel is intimated to be perpendicular (left side” and “right side”) to the line between fields so a couple hundred feet of travel isn’t going to change those angles appreciably. (That might be an interesting exploration, though.)
The problem should ask for the height of the balloon and the horizontal distance to the field or a prospective landing spot so the balloonist can start a steady descent. This all assumes that the balloon does not have a GPS or an altimeter and that none of the passengers has a smartphone ….
Wesley Calvert
December 6, 2010 - 8:12 am -FALSE POSITIVE!
There’s a lot of awful pseudocontext in the world. This is a reasonable enough problem. In particular, the context is not flatly untrue, nor are the operations wholly irrelevant to the context.
Yes, I could make up better problems on this situation. No, this one is not my nominee for the math hall of fame. But it’s not in a class with some of the filth we’ve seen here.
If we’re to stand against pseudocontext, it has to have a boundary. Moreover, the boundary should be distinct from “Is it the best problem you could write about this situation?” Flagging this problem as pseudocontext blurs that boundary.
Don
December 6, 2010 - 12:05 pm -I disagree with Wesley. It’s pseudocontext because it creates a pointless situation and asks a question that the situation doesn’t demand be asked or even seem to make sense. If there’s no reason to ask the question based on the context then it’s just noise. It contributes to why students hate math problems: they do not contribute to understanding, they simply create an exercise in deciphering the variables. If the point isn’t obvious it’s obfuscation for its own sake.
If the question introduced enough other facts that it added value by helping the student learn to cut through the noise to get the pertinent facts that would be one thing. This does not.
Simply altering the problem to be one where the rider took off from one field and is traveling to the other would be all it would take to make the question seem to have some sort of purpose. The total distance between the two fields could be known and the question could be to determine what fraction of the trip has completed. The balloon could be over a landmark where the distance to one field is known but not the other – determine total distance.
Provide time already passed, ask whether the balloon will get them to the next field in time for the 3pm kickoff. Trivial algebra (assuming a constant speed :) but it make the context MEAN something. That’s not an issue of a possibly better-written question, it’s what provides any reason at all for the details.
J Chak
December 6, 2010 - 3:47 pm -This may be getting off-topic, but this could follow a question on figuring out optical rangefinders for golf. (They are based on the premise that hole flags have a fixed height, and they can line the flag up with graduated marks on the lens to determine distance to the flag.) Could use the rangefinder to measure telephone poles to determine their distance … not sure how to make an easy problem out of that though, because angle of the telephone poles complicates things.
Being an engineer, I would try to make an assumption — either that one telephone pole is sufficiently far away to be at a nearly zero-degree angle, or that both poles are roughly equidistant.
Then … aargh! I’m trying to use similar triangles to find the solution, but I’m not sure I have enough known variables.
(Now where the heck did I put that new laser rangefinder…)
jon
December 7, 2010 - 11:32 am -Isn’t finding pseudocontext problems taking the easy way out of this. How about amassing a collection of good problems? That to me seems a much more difficult issue to address. I agree with Wesley. The boundary is smeared here. I will again argue, every textbook bound word problem I have encounted can be considered pseudocontext.
I would really love to see the good ones that already exist.
Dan Meyer
December 7, 2010 - 12:39 pm -To the extent that pseudocontext is a continuous measure (rather than a binary measure) I’ll agree this particular problem isn’t the worst one we’ve featured. But I’m compelled by the fact that a student who knows the actual context of hot air balloons is at more of a disadvantage than a student who is oblivious to hot air balloons, particularly with the height extension.
The student who knows that hot air balloons don’t float a mile in the air will doubt her mathematical analysis – even though it’s correct – simply because the authors didn’t trouble themselves to get the context right. That’s fantastic, frankly. That’s pseudocontext.
Been giving that a shot for the last four years or so.
Here’s a sample homework assignment I grabbed randomly out of the IMP curriculum. There are any number of ways we could improve this problem, but where do you find the pseudocontext here?
Dan Meyer
December 7, 2010 - 2:48 pm -PS. J Chak’s reference to golf course rangefingers just altered my universe. Thanks, bud. Killer suggestion.
Touzel Hansuvadha
October 16, 2011 - 2:14 pm -Dan, the link to the IMP problem is not working. It shows up as a “404 Not Found”. If you can’t fix the link, that’s fine–but I’m very curious to know the name of the assignment.
Dan Meyer
October 16, 2011 - 3:04 pm -That’ll teach me not to download these things when I have the chance. Lost to the wind, I guess.