Let’s celebrate excellence in dead-tree-based math curricula when we see it:
While many texts use those numbered steps to reduce their tasks to something mealy and mushy, this particular text offers useful general advice for solving problems with mathematics. You can almost feel the teacher’s exasperation as the student haggles for hints:
S: I don’t know where to start.
T: Okay, have you made a diagram?
S: Okay now what.
T: Can you tell me about the diagram in words?
S: Okay now what.
T: Have you turned words into labels?
S: Okay –
T: – labels into equations? Have you solved the equation?
That’s decomposition that’ll help the student with problems beyond this one.
Meanwhile, this problem basically cops to the fact that even after it’s decomposed itself into a million mushy little bits, there’s no way anyone has bothered to wade through them all so it says, “C’mon, you guys. Just throw these functions into your graphing calculator and figure out where they cross so we can all get out of here.”
Total knowledge required of projectile motion, speed of sound, the resistance of air, or anything more demanding than how to manipulate Wolfram Alpha?
David WeesAugust 22, 2011 - 9:50 am -
The problem with the first puzzle is that no one would actually use mathematics to solve that. They’d fire up Photoshop (or the equivalent) and start making the brochure & fiddle with the placement of items in the brochure until it worked. It sets up the additional parts of the “puzzle” nicely, but’s not a puzzle worth solving with mathematics.
I think the second problem is more mathematically interesting, but I agree with you that the highlighted piece definitely solves the problem for the students. They are trying to tell a story with the problem, and explain why it is relevant, but fall flat on their face.
Both problems should be kept far away from students you want to interest in mathematics.
AndreiAugust 22, 2011 - 10:44 am -
I find the first problem rather intimidating – a ‘verbal model,’ really?
I’m not really sure what that means, apart from “re-write the question in your own words.”
Damion BethAugust 22, 2011 - 10:58 am -
Except that first problem is EXACTLY what I did as a student in high school making a poster by hand for a foreign language contest. I wanted to lay a ton of flags on a piece of tagboard and wanted so much of an outside margin and new the flag sizes I wanted. I wanted to know how much space to put in between them.
Not everything needs (or wants) to be done on a computer, and I find it a totally applicable problem to give in my math classroom. In fact, I think I may have given that EXACT same problem in my geometry or algebra class in the past couple years.
The second problem does make me cringe, however…
AnnaAugust 22, 2011 - 12:07 pm -
The first problem makes me cringe, too. It’s still feels way too prescriptive. And would you really use algebra to solve that?
And, how about this exchange –
S: I don’t know where to start.
T: Okay, it’s a geometry problem. What do you always do when you don’t know where to start a geometry problem?
luke hodgeAugust 22, 2011 - 12:46 pm -
I wish there were a lot more problems in textbooks like this one that involve drawing and/or interpreting diagrams of easily recognizable situations. However, I am not crazy about the particular road to the solution forced upon the reader. Sure the question is probably in the chapter on solving equations, and it is good to be able to create an equation from a diagram, but it is much simpler and more natural in this case to just look at the diagram and do a couple of computations to figure out the width. That the beauty of being able to create and understand a nice diagram – something that seems complicated becomes common sense. Not saying I haven’t been guilty of things like this as well, asking for unnecessary equations or overly complicated methods, it is often a tough call to make.
Maybe the book could just add another question or two that are more likely to require a variable or an equation. For example, if there were three pictures in the row, the second being 50% wider than the first and third, this may be getting to be a little much for some to juggle in their head without at least labeling the diagram with a variable. Or don’t give the width of the paper and ask them to graph paper width as a function of picture width on the calculator.
Telannia NorfarAugust 22, 2011 - 1:18 pm -
@Damion I understand your comment “Not everything needs (or wants) to be done on a computer” is true but rarely are things done outside of a computer. Before entering into the world of education, I was in publishing. In high school we still designed newspapers and yearbooks by paper, however the real world didn’t. I started my career in publishing in 1994, I never worked in a location where people were doing things by hand.
In general, the textbooks are too helpful but most of the time students don’t even get that far because the reading is dense. My students head for the hills when they see word problems.
Joshua SchmidtAugust 23, 2011 - 9:59 am -
I don’t hate that problem at all either. I find myself immediately thinking (for once) that this is a problem you could simply model as a real life problem. The context of the problem isn’t necessary in the sense that you could just present them visuals without the wording of the problem.
Written problems are great, but I love problems that you don’t need the written context to solve. Fantastic!
Steve PhelpsAugust 24, 2011 - 12:56 pm -
After thinking about this a couple of days, I am not sure I get the difference between the “scaffolding” the two problems are providing.
Both problems concede that the student is just too helpless to solve either on their own. Even the first problem says, “C’mon, you guys. Since you will never figure out what I want you to REALLY do, just follow these five simple steps so we can all get out of here.” Furthermore, I am not at all convinced that these five steps can be applied universally to other problems the students might encounter.
Joshua SchmidtAugust 27, 2011 - 8:46 am -
Steve, I think the main difference between the problems is the first one, from a scaffolding perspective, takes a skill that the student already has and tries to extend it into a relevant problem.
By drawing a picture first, then using labels on the picture, we are teaching a generalized skill that will help the student get through many similar problems. Is it a perfect problem? no. Is it better than most of the ones that my textbook provides me? In my opinion…absolutely. Instead of 5 illogical steps that students wouldn’t get on their own, this is actually the process that I would like them to learn.
Steve PhelpsAugust 27, 2011 - 10:59 am -
Here is what I dislike about the first problem. It puts a nice, tidy box around the problem solving process in such a way that any potential messiness is removed.
Beyond the potential for solving the problem without going through these steps (which I think has been alluded to in other comments), I think the punch line is in Question 17 – “Solve the equation and answer the question.”
The student will need to be reminded to do this, because after the sterilizing effects of questions 13-16, the student will have forgotten all about the original problem.
Greg SchwanbeckAugust 28, 2011 - 8:12 am -
Here’s a way you could make the rock dropping down the well problem more engaging and “less helpful”:
Watch the video “1500 ft Hole” (http://www.youtube.com/watch?v=1JRq80whGDk). Is the claim that the hole is 1500 ft deep accurate? Use your own calculations and the principles of physics to support your answer. A hint to get you started: the speed of sound is about 1125 feet/sec.
BreedeenAugust 28, 2011 - 9:01 am -
I completely disagree. The structure of these problem steps does not “sterilize” the process for students. It provides students with multiple entry points for approaching this problem. The idea that students would have forgotten about what the problem is after engaging in it using visual and geometric strategies speaks to our society’s algebra-centric approach to learning math.
My one tweak to this process would be to remove the ordering. There is no compelling reason that students would need to do these steps in this specific order, though some may find drawing and labeling the diagram a useful first step.