Fermi and the Problem Solved

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.

How can we make problems interesting and challenging?

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!

(For more problems, see here)

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Using Fermi questions in our math classes will remove the pseudo-contexts of much of the problems. They are actually used in real life, unlike many of the questions from a math textbook. Companies use Fermi problems during job interviews as they offer a window into a person’s ability to think on their feet and their creativity. Scientists, economists and engineers use them in their work to get a ballpark idea of the feasibility of their projects. This power of estimation is a key aspect of mathematical thinking. Without it, all the math in the world won’t mean a thing.

Fermi problems emphasize the mathematical processes and help students practise estimation and reasonableness when solving problems. They strengthen number sense, dimensional analysis, and are important in developing a quantitative understanding of the world around us. They allow students to ask the right questions and break down complex problems into smaller, solvable ones. The problems don’t have a definite solution, providing room for interpretation and multiple approaches to problem-solving. The questions are not grade specific and can be used in a range of classes. The open-ended quality of Fermi problems is one of their strengths. 

Encourage intuition in math class through Fermi questions. Give students something meaningful to solve. Tired, old word problems should be a thing of the past.

2 thoughts on “Fermi and the Problem Solved

  1. It was very interesting to read about the Fermi problems. I must admit that this is the first time I heard about it and checked it out.
    The skill of estimation and understanding what a reasonable guess is crucial in real life situations and expands mathematical thinking or should I say, develops mathematical thinking. Giving students opportunities where they can apply math processes by analyzing a situation, making reasonable estimates and reflecting and revising engages and motivates them. When the students have the skill of making reasonable guesses before actually solving a problem, they have a direction and are aware if they are making mistakes and which further helps them self assess and revise. This builds confidence not only in Math but other areas where problem solving is required.

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  2. David, thanks for your post. This past year, our school took up the initiative of Fermi questions for our students once a week. During our professional learning time (home room teacher would be on a rotating schedule when they would work on something of their interest) a teacher would come into the class to cover and present a Fermi question. This was a challenge for some teachers, and unfortunately, not all teachers participated in this student learning opportunity (although it was “mandated” by Admin). What I really liked about the Fermi questions was the giganticness of the question, and to see how students approached it. At first, students in the grade 5 class that I covered were freaking out. “We can’t do this! This question is impossible!” When posing to them one simple question, such as “what kind of math thinking do you think you’ll need to use?” It seemed to settle them and they were able to approach the task and you could see their wheels turning. I think the importance of asking these types of questions is useful and helpful on many different levels, not just for thinking mathematically.
    I’m not sure if our school will continue Fermi questions next year, but I think it’s something that I’d like to do in my home room.

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