This Week’s Installment
What mathematical skill is the textbook trying to teach with this image?
(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)
Team Me: 5
Team Commenters: 3
Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”
I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.
After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.
(See the rationale for this exercise.)
This was a nail-biter between Team Commenters and Team Me this week, with Team Commenters narrowly tipping the scales in their favor.
The judges rule that this satisfies the second rule of pseudocontext:
Given a question, the assigned method isn’t a method most human beings would use to find it.
Reasonable people might wonder about the dimensions of a water tank. The judges rule that most human beings would use a tape or a stick or any other kind of measuring device to answer it, not a cubic polynomial.
I can’t think of any way to neutralize this pseudocontext. The number of actual contexts for cubic polynomials with non-zero quadratic and linear terms is vanishingly small.
Here is an activity I would much prefer to use to teach the construction of polynomials. It doesn’t involve the real world but it does ask students to do real work.
One motif in pseudocontextual questions seems to be treating as a variable things that, you know, don’t vary. I have a funny video playing in my mind of some surprised fish watching the volume of their tank become negative. But happily the volume of that tank is not varying, inasmuch as it’s sides are made of glass.
Joel PattersonDecember 17, 2016 - 7:43 am -
You know, Dan, you should take a more positive perspective on math textbooks and see them as interdisciplinary works connecting mathematics with surrealist artistic sensibilities.
Ivy KongDecember 17, 2016 - 12:17 pm -
There is no way it has anything to do with triangles, so my vote is on the congruent triangles option. The volume of the tank could be disguised as a polynomial, and we could figure out the mean, median and mode of the types of fish…oh wait, the logs in the tank kind of form a couple triangles…
Max GoldsteinDecember 17, 2016 - 4:19 pm -
Well the volume of the tank is the obvious context, and maybe something about the light distribution. I could see descriptive statistics for average amount of food, fish species or lifespan, and so on.
I’m going to go with polynomials, since that sounds like it could be shoehorned in. Triangles seem too remote even for psuedocontext.
Robert KaplinskyDecember 18, 2016 - 7:57 am -
Like Max, I’m guessing it isn’t volume. I haven’t a clue how it could have anything to do with polynomial roots. Congruent triangles seems trivial here. So, I voted for mean, median, and mode… perhaps with water levels.
DanDecember 18, 2016 - 1:45 pm -
There are (possibly not very natural) ways to use all of the options provided. An open rectangular prism necessarily involves shapes that can be resolved into the sum of congruent triangles. Why anyone would do that is a different story! Certainly not for its usefulness in real life. Someone else has already mentioned that statistical measures of centre might be useful in surveying any number of things in the fish tank, from food levels to water levels over time. Also the volume of water in the tank can be described as a polynomial, which when factorised provides the usual volume formula for a rectangular prism as a function of some length parameter x. Then the roots have a definite meaning with respect to that length parameter. Why one would do this in real life is unclear, although it’s not difficult to imagine some kind of manufacturing setting where optimal design of the tank is required to be based on some cost functional involving length as parameter as parameterised by the same x. Or maybe it’s just a lead into calculus based optimisation problems…it has a scent of real world application but is manageable for students seeing this stuff for the first time.
I voted polynomial roots because I think that’s the most likely content area for which the picture might be construed as involving meaningful mathematics! Although, in truth, having a physical application ought not be the delineating factor for maths to be meaningful. For example, mathematics gives me pleasure just for its own sake and teaches me to reason about useless things as much as useful things, so it’s meaningful to me even when there is no obvious application to the real world! But anyway!
Dan MeyerDecember 18, 2016 - 4:09 pm -
No one will ever find me making the counterargument, that math must have a physical application to be meaningful. The implicit argument of this series is that if you’re going to shoot for a physical application, you’d better not miss.
Bowen KerinsDecember 18, 2016 - 7:40 pm -
By the way, question (c) isn’t even answerable. The factoring gives (x-1)(x-2)(x+5), and then we’re told the volume is 3 ft^3. So … does that mean solving (x-1)(x-2)(x+5) = 3, then picking off the three dimensions from the pieces (x-1), (x-2), and (x+5)?
NO, because they told us the length is x! That means the other dimensions are … uhh … god how broken and stupid is this.
The legit context for this would be Snell’s Law and the actual location of a fish relative to the position we think it is.
Max GoldsteinDecember 18, 2016 - 8:36 pm -
What’s maddening is that polynomial is completely arbitrary. It doesn’t relate to a property of the tank in the way that, say, SA = 2ab + 2ac + 2bc does. (If there IS some useful interpretation of the factored version, it would be really useful to point that out!)
William CareyDecember 19, 2016 - 5:07 am -
One motif in pseudocontextual questions seems to be treating as a variable things that, you know, *don’t vary*. I have a funny video playing in my mind of some surprised fish watching the volume of their tank become negative. But happily the volume of that tank is not varying, inasmuch as it’s sides are made of glass.
Dan MeyerDecember 19, 2016 - 7:17 pm -
Featured this one up top.
TrevorDecember 19, 2016 - 11:02 am -
Hey everyone, I am a pre-service teacher in Michigan, and I found this blog from a list of recommended edubloggers. I just wanted to hop on the comments and tell you how awesome I think this thread/game is.
I am relatively new to teaching (I have taught chemistry in the past and have tutored a lot of high school math), but I am glad to see that I am not the only one who feels that “real world examples” of math problems miss the mark.
I am curious you all write your own “real world” problems that are more reasonable, or if you discuss this at all with your students? As a tutor, I pointed it out when the student was frustrated with the problem, but if the student didn’t struggle, I let them do the problem without interruption.
Dan MeyerDecember 19, 2016 - 7:31 pm -
Hi Trevor, thanks for the comment. Generally when it comes to application problems, I try to inoculate pseudocontext by asking myself, “Given this context, would a normal human being ask wonder this (mathematical) question?” And if so, then, “Given this question, would a normal human being use this (mathematical) method to answer it?” If the answer is yes for both, then we have a promising task.
Anna schollDecember 19, 2016 - 5:52 pm -
I voted polynomials thinking about swimming paths of fish.
In part because I had students use an aquarium theme for a functions project and that was their polynomial context.