Gee, That’s the First Time I’ve Heard That, Dept.

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.

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5 thoughts on “Gee, That’s the First Time I’ve Heard That, Dept.

  1. “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.”

    So they all must be getting A’s!

    My son did his share of Fermi problems. They are cute and based on memorizing simple facts and using simple math. A steady curriculum of Fermi problems would have done little to improve his results in college. However, the traditional AP Calculus track did help, and he now successfully handles proofs in classes like abstract algebra.

    This is a classic case of a modern educational pedagogue trying to claim the higher ground of better math understanding for all students while showing absolutely no evidence of a specific longitudinal curriculum success.

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  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!”

    This guy should get out of his classroom and learn more about the “real life” of engineers. They do the calculations exactly and then add on safety factors. The apply fatigue S-N data. They may apply rough estimates of beam theory to many different problems, but learning estimates don’t come first. We applied beam theory to large ships, but we also had three levels of detailed calculations to get to that point. Estimates aren’t more real than the detailed calculations. They are less real. You have to take the governing equation and decide which variables are primary and which are second-order. That’s what Fermi did for the nuclear bomb test. Estimating the number of piano tuners who can dance on the head of a pin is not real life.

    On one test I had in college, we were told about two fishermen sitting at the shore who timed the period of the swells coming in. The question was to determine how far out at sea the center of the storm was. The answer had to come from the governing equation with assumptions about time and long distances. I mentioned that problem to that old (traditional) professor recently and he said that was an easy one. It was, but you had to know the governing equation and which variables to ignore or assume some value. Imagine! we had that on a traditional test.

    We also had questions like this in our real world of traditional education. Given a big tank partially filled with water and a boat inside. You are sitting in the boat and decide to throw out your heavy anchor to the bottom of the tank. Does the level of the water in the tank go up or down?

    It’s surprising to hear educational pedagogues come up with problems like these as if they are some grand new epiphany of learning. Worse, they want to use those problems as the center-pieces of in-class group learning without worrying about the details of individual discovery, mastery of skills, and any sort of transference. No. you first have to understand and master the governing equations. THEN, you can do transference. Duh! Double duh!

    For problems with no one single answer, you have to learn the mathematical techniques of optimization and be able to define a proper merit function. There are NO problems that have multiple answers. You have to quantify the unknown and not leave it to some random whim or opinion at the end. The merit function might have weights that are based on opinion, but they are calibrated exactly for everyone to evaluate and change.

    It’s quite amazing to see educators read anecdotes they don’t understand to validate their hypotheses. Real scientists work very hard to try to disprove their theories. Otherwise, they end up looking very silly. But what does peer review mean in the education world when they all learned the same things by rote and claim that their turf is more important than mere facts and rote skills. They claim to know what the real world is all about. Meanwhile, the high school world goes on with their traditional AP-track math classes and the real world pushes proper curricula back to 7th grade. Skills tracking is hidden at home, the academic gap increases, and the only un-reality left is K-6 education.

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  3. The inanity is endless. Criticize everything for simplistic reasons and then replace with something that is really nothing more than a figment of someone’s imagination.And finally spend millions of dollars with no accountability in sight. Repeat.

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  4. I wonder if our blogger has ever had to mark Fermi problems. Because I have, and they are beyond painful.

    You can’t assume a correct answer is done correctly, because it might have cancelling poor assumptions. Good logic might be undone by incorrect original data, so a wrong answer might score well. The result is that every single calculation and assumption must be checked — and since everyone has slightly different working you check each one from the beginning. It’s one thing for Google to have to listen to some prospect show his working in an interview, but it’s another to work through 30 of them over five or six hours with enough detail to give marks.

    And when done by 15 year-olds you end up guessing what they did and why they did it unless they are naturally good at explaining.

    Fermi problems are good to do in a Maths classroom as an interactive exercise to encourage open minded thinking. As an actual whole class exercise they suck badly.

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