At a math blog, I came across the following paragraph:
“Word problems in math textbooks often give much of the information needed to solve them. There is no mystery. Students walk away from a math course with the only skill acquired being the ability to decode the textbook. They are just swapping numbers and plugging in different information. As a result, the so-called problems are no longer problems. They are routine and predictable. The problems are too scaffolded and the students realize that it’s an exercise in futility. An insult to their intelligence. While practice is indeed a fundamental part of math, when problems are variations of the same one, the motivation to complete them is lost. They don’t see the point of it all.”
This is the standard complaint levied against the typical word problems one sees in math textbooks. Well sorry to disagree with this grand master, but I’ve been using Dolciani’s algebra book for my 8th grade algebra class and believe me, the students are not finding the problems predictable. Dolciani, like other good math book authors, does vary the problems so that one is not working the same problem over and over, which is the standard complaint–i.e., it’s just plug and chug.
But this blogger doesn’t even like such variations and scaffolding. Sorry, but that’s how you understand and master the basic skills in math–by getting students to see the structure of problems and how to solve them through the initial worked examples, and then stretch their capability and extend the problem solving principle to situations that are just a bit different, and more difficult.
No, what this blooger likes are what are called “Fermi problems”
Look to Enrico Fermi. The Italian physicist had a gift for making accurate estimates of seemingly unsolvable questions. Given little information, he was able to provide educated guesses that came very close to the actual answer. His most famous question, “how many piano tuners are in Chicago?” seems to make no sense, but through a series of questions, estimations and assumptions, he arrived at a reasonable answer. Legend says that Fermi calculated the power of an atomic explosion by looking at the distance his handkerchief travelled when he dropped it as the shockwave passed. He determined it within a factor of 2. For a discipline that is always looking for realistic applications, math class would do well to use Fermi problems. It doesn’t get more real-life than that!
While there’s nothing wrong with such problems per se, they should not be used as a starting point or replacement for learning math nor as the fundamental definition of what math is used for. Fermi problems are percent/scale-up problems. How many golf courses are there in the US? How many molecules are in a mountain? These are classic IQ and job interview types of questions. The fallacy is that one cannot memorize (!) a lot of facts to make these estimates easier. You can practice these problems to get better at them. They can convince others that you are a genius. A big fact to memorize is how many people are in the US. A second one is how many people are in your state. Then estimate the number of golf courses (or whatever) in your state and scale it up. Often, when someone asks you one of these questions, they are happy if you can come up with some reasonable process for estimation.
A steady diet of these things does not teach students general and transferable problem solving skills that they will need in other math courses. The belief seems to be “Give them top down type problems that force them to learn things on a ‘just in time’ basis, as if there is a problem solving schema that will emerge, given enough time and enough off the wall problems. Solving Fermi problems depends on memorizing simple facts and using simple math. It is not what math is all about.
Talk to the parents of the students who are on the track to AP calculus and on to STEM majors. They solved lots of the traditional word problems people like this blogger hold in disdain. Dolciani’s algebra books didn’t skimp on problems. Every chapter had word problems tailored to the particular math skill that was the focus of that chapter. If the chapter was on algebraic fractions, then the rate/distance problems and mixture problems given in that chapter relied on knowledge of algebraic fractions to set them up and solve.
My students are finding the problems challenging. It took some time before we were at the point where a certain type of problem was familiar, and for me to then up the ante. But unfortunately, the beliefs espoused by this blogger are very typical and people who teach the traditional problems are viewed as doing their students a disservice.
A math professor I know has this to say about teaching students how to solve problems using things such as Fermi problems:
“I WANT my students to look at the sophisticated work I give them and say “Huh, this is no problem I just do such and such and so and so, and this will get me to the answer”. It is the students who have to struggle and fret over straightforward stuff that I worry about. Why do they insist on making easy things hard and putting roadblocks in students’ way?”
But what does he know? He’s just a mathematician who happens to think like one.