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.
Providing guidance
Whether your students work in groups or individually, many will ask for guidance while doing the tasks. The amount of guidance that students need should decline as they become familiar with this type of problem. Early in class, you are likely to provide guidance in the form of questions directly related to each stage of the solution process:
- What do you know that is relevant?
- What assumptions can you make?
- How plausible are your assumptions?
- Is your chain of reasoning accurate?
- Can you do the problem another way and see if the result is the same?
- In your answer, do you spell out your assumptions, reasoning, solution, and checking procedure clearly?
The amount and type of help you provide the students depends upon your goals for the task. Later, if your primary goal is to encourage students to struggle with solving the problems on their own (and learn that they can "do estimation"), you may choose to provide very little assistance.
Reporting out of individual or group work
If you decide to come together as a large group to discuss what students came up with (or report out), it is again helpful to decide the degree to which you will participate in these discussions (and will depend upon your goals for the session). For instance, you can facilitate the students' discussion by having them defend their ideas and write their ideas on the board, and adding almost none of your own. This approach of focussing on critical questions can direct students away from viewing you as the authority. Alternatively, you might model your own estimation skills for students by leading the discussion, soliciting student comments and organizing them in a useful manner and adding comments to guide them into an understanding of the problem.
Formal and informal use
These tasks can be used formally or informally. In formal assessment (where you grade the assignment as an examination), do not intervene except where specified. Even modest interventions - reinterpreting instructions, suggesting ways to begin, offering prompts when students appear to be stuck - have the potential to alter the task for the student significantly.
In informal assessment (an exercise, graded or non-graded), you may want to be less rigid in giving the students help. Under these circumstances, you may reasonably decide to do some coaching, talk with students as they work on the task, or pose questions when they seem to get stuck. In these instances you may be using the tasks for informal assessments - observing what strategies students favor, what kinds of questions they ask, what they seem to understand and what they are struggling with, and what kinds of prompts get them unstuck. This can be extremely useful information in helping you make ongoing instructional and assessment decisions. As students have more experiences with these kinds of tasks, the amount of coaching you do should decline and students should rely less on this kind of assistance. Evidence that students are learning from these activities comes in two different forms. First, the quality of the solutions they produce will improve. Second, they will use the key questions (are the assumptions reasonable? is the logic correct? is the answer plausible?) as they try to estimate a solution.
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
- Prepare by reading through the 'Plausible Estimation' task on your own and coming up with your own solutions.
- Hand out copies of the task to students, either working individually or in groups.
- 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.
- Walk around and listen to students as they discuss and work through the problems, providing guidance as necessary.
- Have students present their solutions, either in written or verbal form.
Tell me more about this technique:
Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving ||
Reasoning from Evidence
