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.

• 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?

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.

Teaching Goals

• The student learns modeling methods appropriate for the subject
• The student learns to evaluate methods and materials of this subject
• The student generates many potential solutions to a given problem
• The student applies principles and generalizations to new problems and situations
• The student analyzes problems from different points of view

Suggestions for Use
Introducing 'Plausible Estimation' tasks for the first time
Many students will be unfamiliar with the open-ended nature of 'Plausible Estimation' tasks, and you may experience some resistance from your students. You can reduce this in three ways.

• First, prepare your students by telling them that the goal of these tasks is to get them thinking like a mathematician (or an astronomer, etc). You are looking for their ability to make estimates in situations where estimation seems difficult, by following simple assumptions and straightforward reasoning to arrive at a reasonable solution. You may need to "sell" these tasks to the students since many will be unaccustomed to open-ended problems that don't assess manipulation of formulas. You might point out that estimation is a key element in everyday life, using examples such as the following:
  • In business, a typical problem might be, "How big is the market for mobile phones?"
  • In educational policy a question might be, "If the School District is going to reduce class size to 25 students, what will be the implications for local taxes?"
  • In preventive medicine, a typical problem might be, "This winter, everyone over the age of 60 years in New York City needs to be vaccinated against influenza - how fast can this be done?"
  • On a personal level it might be, 'I'm going to work in fast food place to cover my college fees - how much study time will this leave?"

• Second, to help your students adjust to these types of problems, the first experiences with 'Plausible Estimation' tasks should be via non-graded, in-class, group-based exercises. Since the tasks are non-graded, students can work without fear of "messing up their grade." And, if tasks are done in-class, students can receive help from you as they work through the exercise. Finally, if the task is group-based, then the students can struggle together and receive support from one another.

• Finally, students will be anxious to know what a "good" answer is. You can provide them with various rubrics (see analysis section of this document) that describe the kinds of answers you expect to see and examples of each (or at least of a good answer). Knowing what a good answer looks like is itself an important learning goal.

Group work versus individual work
The open-ended nature of these tasks makes for great group work problems. Students can discuss various measures and their merit and are likely to come up with many more ideas than if they worked alone. The CL-1 Collaborative Learning site can provide instructions on how to use group work effectively within the classroom. However, individual work may give you more clues as to each student's sophistication with this type of problem.

Presumed background knowledge
One nice attribute of 'Plausible Estimation' tasks is that they require almost no mathematical knowledge. Students do need to have a basic knowledge of geometry concepts (area, perimeter, length), basic numeric skills (multiplication, addition, subtraction, and division, using large numbers), and an understanding of units and conversion of units.

Step-by-Step Instructions

1. Prepare by reading through the 'Plausible Estimation' task on your own and coming up with your own solutions.
2. Hand out copies of the task to students, either working individually or in groups.
3. State your goals for the 'Plausible Estimation' task, emphasizing that they should be able to defend both their assumptions and the reasoning which leads to their answer.
4. Walk around and listen to students as they discuss and work through the problems, providing guidance as necessary.
5. Have students present their solutions, either in written or verbal form.

Variations
The tasks included in this site can be downloaded and used without modification. If you choose to develop your own 'Plausible Estimation' task, you can follow the pattern used in these tools.

• First, select a question whose quantitative answer you can look up in a reference manual (as a check against the students' answers), and which can be derived from fairly simple assumptions or estimates. If the solution process needs a little-known estimate, provide that to the students.

• Then, ask the students to estimate the value, placing an emphasis on their assumptions and chain of reasoning.

The following examples from various disciplines may provide some insight into creating your own variations.

Analysis
Student work can be measured against three criteria:

• students can make sensible assumptions relevant to the task;

• students can design a computational strategy that logically estimates the desired quantity from the given quantities; and,

• students can ascertain the reasonableness of the computed estimate and communicate the assumptions upon which the estimate was based.

Malcolm Swan
Mathematics Education
University of Nottingham
Malcolm.Swan@nottingham.ac.uk

Most assessment practices seem to emphasise the reproduction of imitative, standardised techniques. I want something different for my students. I want them to become mathematicians - not rehearse and reproduce bits of mathematics.

I use the five 'mathematical thinking' tasks to stimulate discussion between students. They share solutions, argue in more logical, reasoned ways and begin to see mathematics as a powerful, creative subject to which they can contribute. Its much more fun to try to think and reach solutions collaboratively. Assessment doesn't have to be an isolated, threatening business.

Not just answers, but approaches.

Malcolm Swan is a lecturer in Mathematics Education at University of Nottingham and is a leading designer on the MARS team. His research interests lie in the design of teaching and assessment. He has worked for many years on research and development projects concerning diagnostic teaching (including ways of using misconceptions to promote long term learning), reflection and metacognition and the assessment of problem solving. For five years he was Chief Examiner for one of the largest examination boards in England. He is also interested in teacher development and has produced many courses and resources for the inservice training of teachers.

Jim Ridgway
School of Education
University of Durham
Jim.Ridgway@durham.ac.uk

Thinking mathematically is about developing habits of mind that are always there when you need them - not in a book you can look up later.

For me, a big part of education is about helping students develop uncommon common sense. I want students to develop ways of thinking that cross boundaries - between courses, and between mathematics and daily life.

People should be able to tackle new problems with some confidence - not with a sinking feeling 'we didn't do that yet'. I wanted to share a range of big ideas concerned with understanding complex situations, reasoning from evidence, and judging the likely success of possible solutions before they were tried out. One problem I had is that my students seemed to learn things in 'boxes' that were only opened at exam time. Thinking mathematically is about developing habits of mind that are always there when you need them - not in a book you can look up later.

You can tell the teaching is working when mathematical thinking becomes part of everyday thinking. Sometimes it is evidence that the ideas have become part of the mental toolkit used in class - 'lets do a Fermi [make a plausible estimate] on it'. Sometimes it comes out as an anecdote. On graduate told me a story of how my course got him into trouble. He was talking with a senior clinician about the incidence of a problem in child development, and the need to employ more psychologists to address it. He 'did a Fermi' on the number of cases (wildly overestimated) and the resource implications (impossible in the circumstances). He said there was a silence in the group...you just don't teach the boss how to suck eggs, even when he isn't very good at it. He laughed.

Jim Ridgway is Professor of Education at the University of Durham, and leads the MARS team there. Jim's background is in applied cognitive psychology. As well as kindergarten to college level one assessment, his interests include the uses of computers in schools, fostering and testing higher order skills, and the study of change. His work on assessment is diverse, and includes, the selection of fast jet pilots, and cognitive analyses of the processes of task design. In MARS he has special responsibility for data analysis and psychometric issues, and for the CL-1 work.

About MARS
The Mathematics Assessment Resource Service, MARS, offers a range of services and materials in support of the implementation of balanced performance assessment in mathematics across the age range K to CL-1. MARS is funded by the US National Science Foundation, and builds on earlier funding which began in 1992 for the Balanced Assessment Project (BA) from which MARS grew.

MARS offers effective support in:

The Design of Assessment Systems: assessment systems are tailored to the needs of specific clients. Design ranges from the contribution of individual tasks, through to full scale collaborative work on test development, scoring and reporting. Clients include Cities, States, and groups concerned with educational effectiveness, such as curriculum projects and professional development initiatives.

Professional Development for Teachers: most teachers need help in preparing their students for the much wider range of task types that balanced performance assessment involves. MARS offers professional development workshops for district leadership and 'mentor teachers', built on materials that are effective when used later by such leaders with their colleagues in school.

Developing Design Skills: many clients have good reasons to develop their own assessment, either for individual student assessment or for system monitoring. Doing this well is a challenge. MARS works with design teams in both design consultancy and the further development of the team's own design skills.

To support its design team, MARS has developed a database, now with around 1000 interesting tasks across the age range, on which designers can draw, modify or build, to fit any particular design challenge.

Introduction
Description, Purpose, and Limits
Goals, Use, and Instructions
Variations and Analysis
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