How Many Piano Players: Fermi Questions in an 8th Grade Class
By Glenn Kenyon
The San Francisco School
The longer I teach 8th, the more comfortable I am that a large majority of my students are not only ready for the algebra, but also find relief in it. A student once wrote, “algebra is almost meditative once you learn the procedures to solve for x or graph a line”. It seemed an odd statement until I considered his history with math. As his seventh grade teacher I saw him struggle with topics such as scientific notation, ratios and geometric proportionality. Regardless of the problem solving nature of our math curriculum, there was always something mysterious about how numbers related to each other that frustrated him.
It has become a pattern that some students who do relatively poorly with these topics in my seventh grade class go on to be quite successful in my 8th grade algebra class. In part it could be the various developmental milestones through which they are passing. It could also be systemic in that 8th grade carries a significant weight in our students’ minds as they look at what high school has to offer them. But what I really think is going on is that some students have never come to terms with the numbers they have been manipulating since they started school.
For years I have asked my students to build the number “one million” from rainbow cubes and paper, measure scaled versions of our solar system on the playground and study images of powers of ten from the microscopic quark to the universe. We have also learned the rules of exponents, calculated the distance from Mars to Jupiter and determined what each of us owes as a portion of the national debt. In all cases, I was directing the problem, the procedure and even the solution. As creative or real-world based these activities may have been, there were students who dutifully followed the task without grasping the numbers and their implications.
Without a doubt, middle school math needs to prepare students to deal with quantitative situations in their lives outside school. The NCTM standards state that students need to “develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation.” They need to “determine which data are appropriate for their needs, to understand how the data were gathered at the source, and to consider limitations that could affect interpretation.” The question is: How do we best address this?
One piece of the puzzle for me has been using Fermi Questions with my students. So much of the focus of 8th grade is algebra that number sense takes a distinct back seat. In addition, our students have graphing calculators. Even when number problems arise that could be solved mentally, they often resort to technology. As a result, I have students who used to know their multiplication tables now using their calculators to multiply 7 and 8! Fermi questions let us put down the algebra books, algebra tiles and graphing calculators and do a little “back of the envelope” estimating.
What are Fermi Questions? They receive their name from Enrico Fermi, an Italian physicist known for his participation in the Los Alamos atomic bomb project. He is known for having estimated the power of an atomic bomb detonation by letting a hankerchief drop as the pressure wave passed over him in his observation point. (He died at age 53 from stomach cancer.)
Fermi questions emphasize estimation, numerical reasoning, communicating in mathematics, and questioning skills. Students often believe that "word problems" have one exact answer and that the answer is derived in a unique manner. Fermi questions encourage multiple approaches, emphasize process rather than "the answer", and promote non-traditional problem solving strategies.
I was intrigued how my students would react to a Fermi problem. After giving them a brief background of his life, I presented them with a classic Fermi Question: How many piano tuners are there in our city (San Francisco). They looked at me perplexed? What were the numbers they were expected to work with? I told them we had no specific numbers to work with other than those we might know of as a group, such as the population of San Francisco (approximately 800,000). One student raised her hand. “I actually know that there are about 15 piano tuners in San Francisco because we have a piano and we looked for a tuner once”. There was rumbling in the class and another student said that he didn’t believe that number was valid because some people advertise from other cities in San Francisco’s Yellow Pages. It hadn’t occurred to me to research the actual number before the activity, but I suggested that we try as a group to work out a reasonable estimate. The unexpected debate about the validity of a number made for an enthusiastic discussion.
I must admit that I entered the class with little more than a question, a personal comfort level with large numbers and a fascination for the historical context of math discoveries and inventions. I was lacking many concrete numbers, specific lesson plan structures or worksheets, or any appreciation of what a piano tuner actually does. I would normally consider this a poor teaching practice that causes chaos and terrible learning outcomes. But in this one instance, my lack of knowledge was advantageous. My students were engaged, thoughtful, and some were even quite knowledgeable about piano tuners.
As I facilitated the ensuing conversation, I found myself categorizing our assumptions into two groups: what we knew as near fact and what were our best guess assumptions. These categories later evolved into “Fact” and “Assumption”. In both categories, there were quantitative facts (population of San Francisco) and assumptions (a piano tuner needs two days to tune a large piano). There were also other, non-qualitative facts and assumptions. For example, we were pretty sure that not every owner of a piano would actually hire a piano tuner in a given year. Another assumption is that homeowners were more likely to own a piano than renters. We tried to use these assumptions to guide our quantitative work, but there were a wide range of opinions. In addition, many of my students approach numbers as static entities, to the point where rounding a number from, say 18,000 to 20,000 was motive for debate. Since my goal was to arrive at a number by the end of the 50-minute period, I made several decisions along the way to impose my understanding on the problem if I felt we would be debating too many fine points along the way.
By the end of that math period, we had estimated approximately 100 piano tuners in San Francisco. The number seemed quite large given all of our assumptions, so we spent about 5 minutes looking over our numbers. Fermi problems often deal with very large numbers and choices about magnitude play very important roles in the procedure. As it turned out, we looked over our numbers and felt that we had overestimated by a factor of 10 the number of households that were capable of having pianos. Decreasing that number ten-fold brought our number of piano tuners down to ten, thereby essentially agreeing with the girl who started the debate in the first place and the boy who was sure her answer was too high. The fact is that given what we knew, both of their answers were very close to each other.
Class ended and I was left thinking of the importance of what had just happened. My students were highly engaged in solving a problem. They had several opinions about the size and validity of our estimates, which meant to me that they were grappling with the real meaning of numbers in context. Our math class felt more like a humanities or science class, replete with debates, hypotheses and consensus-driven decision making. I wanted to do more such problems and yet, wondered how many times I could bring such a question to class with little preparation and have such positive learning outcomes.
Nothing ventured is nothing gained, so the following week I came upon a question more organic to our preschool to 8th grade school: If we could weigh the entire student body on a given day, what would the result be in pounds? I invited our head of school to observe the class. This time around I was sure that I did not want to lead the conversation, but rather, I would have them work in groups of four students. Once again, I seemed to have hit upon a good question because the students immediately started working. I gave each group a large piece of paper, a marker and the instruction to list all their facts and assumptions in a t-table as well as show all their calculations. They wanted calculators, but I said that technology would defeat the real purpose of this type of calculation. So while they missed the comfort of their calculators, they were obliged to make their numerical assumptions rounder and more friendly to paper and pencil work.
The two adults in the room wandered from group to group alternately observing and offering advice when it was requested. The problem had an easy data entry point, namely the number of students in their own 8th grade class and the approximate weight of a “normal” 8th grader. Nearly all the groups worked from that natural starting point and went down through the grades. What fascinated me was the assumptions of average weight than they groups came to independently. My students demonstrated a clear concept of what “average” looks like in the real world. Many of these same students would be challenged to calculate averages accurately from a group of numbers given to them. Nearly all the groups assumed about a 10 pound weight difference from grade to grade, which, when applied all the way down to 3 year olds in our preschool, came out to about 40 pounds (this actually coincided with my own son’s weight at that age). They also rounded out the number of students in each class to 30 in the middle school, 20 in the elementary and preschools, which while undercutting the whole school by about 10%, actually allowed them to make the calculations relatively quickly and effortlessly.
With little guidance or modeling aside from the piano tuner problem the previous week, the 8th graders were able to work out the intricacies of this problem with little controversy. There were 7 groups of students working on the problem for about 40 minutes and when I called on each group to give their answers, I was shocked to see the degree of agreement of the numbers. All the answer came in between 35 and 40 thousand pounds. I converted their answer to tons: between 18 and 20 tons. The students were surprised when I told them that these answers were remarkably uniform given the nature of the problem. To them, the answers were very distinct from each other. I explained that in this type of problem where assumptions are crucial, coming within 10% of each other was amazing. They shared their posters with each other and concluded that, indeed, while their assumptions varied from group to group, they seemed to “average” out in the end.
Later that week I contemplated what my next step should be. I was very enthusiastic about the outcomes of the two classes I dedicated to Fermi Questions. I felt I had come upon some good organizational strategies with the fact and assumption t-table. Pedagogically I knew that moving away from calculators would help some of my students recover some latent number sense that was being supplanted by technology. All these outcomes were directly supporting the NCTM standard of developing number sense in a real-world context. I found I was personally influenced by the activity to the point that I looked at the school, the city and beyond as sources of intriguing Fermi Questions. I drew up a list and started planning out a year long unit and then I stopped. Was I on the verge of formulating the solution to my students’ number sense woes? Or rather, was I going to kill the magic of these last two classes by institutionalizing the format, the types of questions and the acceptable responses?
I believe that answer lies somewhere in-between these two extremes. It is like so much of teaching, a delicate balance between art and science. There is no bullet proof lesson to ensure student success and yes, even the best lesson can be a disaster if delivered poorly. I came to see that if I started assigning Fermi Questions on a weekly basis I would drive the novelty into the ground. I would have troubles finding the right questions. In fact, a few weeks later, I tried another question: how much turkey was consumed in the US on Thanksgiving? I had the approximate answer and the students worked on the problem, but there was a lack of spirit and innovation in the class that day. The luster was gone and all had returned to routine. Routine is great for many aspects of class, but not for Fermi Questions.
In the future, I will continue to view our world as a series of Fermi Questions. When I sense the moment is right, I will spring more of them on my students and share with them the wonders of our quantitative existence.