CL-1: Field-tested Learning Assessment Guide (FLAG): CATs: 'Plausible Estimation' Tasks

Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving || Reasoning from Evidence

Classroom Assessment Techniques
'Plausible Estimation' Tasks

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Description
These tasks (an example) are sometimes called 'Fermi' problems after the physicist Enrico Fermi (1901-1954). One favorite problem was, " How many piano tuners are there in Chicago?" Fermi problems have the following characteristics:

An interesting estimation problem is posed in a simple way.

Most people instantly respond by saying that it is a problem they could not possibly solve without recourse to reference material.

An estimate of the solution may be found by a series of simple steps that use only common sense and numbers that are either generally known or are amenable to estimation.

Thus, one way we could estimate an answer to Fermi's question about how many piano tuners are in Chicago is to:

estimate the size of the population
estimate the number of households in the population
estimate the total number of pianos in one's own class, family, street, church etc.
estimate the frequency of tuning
estimate the time it takes to tune a piano
estimate the number of piano tuners.

The downloadable materials for students begin with the sample task and solution that appears below. When we use these tasks for assessment, we are looking for:

sensible assumptions
careful reasoning which is carefully communicated, and
sensible use of units.

Example of a Task and Solution
Plausible estimation tasks are designed to see how well you can develop a chain of reasoning that will enable you to estimate an unknown quantity from things that you already know or can easily guess at. The best way of explaining this is to give an example.

How much will you drink in your lifetime?

How many baths would this fill?

My first reaction might be "How on earth can I answer that?" The secret is not to panic, but to think about what I do know. I shall start by writing down a few assumptions

Assumptions
Today, I had 4 mugs of coffee (about one and one half pints) two glasses of orange juice (half a pint), a can of soda (about half a pint), some milk on my cereal (about a third of a pint). I must have missed something...

So I shall write down an assumption:

On a typical day I drink about 3 pints.

Now I also need to know about bathtubs. I am 6 feet tall, and when I soak in the bathtub, I can reach the taps with my toes, while keeping my head above water, so the bath must be about 5 feet long. Its about 2 feet 6 inches wide, and about 1 foot deep. Here then is my second assumption:

A full bath holds about 5 x 2.5 x 1 = 12.5 cubic feet.

One last assumption:

I will live about 75 years.

Calculations.
The units at the moment are incompatible. I've got pints and cubic feet.
This is where I need a reference book.
It says that: 1 US pint = 29 cubic inches
I know that 1 cubic foot = 12 x 12 x 12 = 1728 cubic inches. (12 inches are in a foot)

So, lets change all the units to cubic inches:
I drink about 3 x 29 = 90 cubic inches per day, (approx).
In 75 years that is 90 x 365 x 75 = 2,500,000 cubic inches (approx)
My bath holds 12.5 x 1728 = 22,000 cubic inches (approx)
So that means I will drink about 2,500,000 ÷ 22,000 = 113 bathfuls.

Answer: In a lifetime I will drink a little over 100 bathfuls.

Assessment Purposes
There are three assessment purposes:

to see how well students are able to make reasonable estimates of everyday quantities;

to see how well students are able to develop a chain of reasoning that will arrive at an estimate of the desired quantity from given quantities; and

to see how well the student can ascertain the reasonableness of the estimate and communicate the assumptions upon which the estimate is based.

Limitations
The 'Plausible Estimation' tasks do not assess specific kinds of mathematical knowledge; rather, they assess a student's mathematical thinking skills. However, these tasks do pick up poor skills in arithmetic, handling large numbers, and conversion of units.