Suggestions for Use
Introducing "Creating Measures" tasks for the first time
Many students will be unfamiliar with the open-ended nature of 'Creating Measures' tasks, and you may receive some initial resistance from your students about this. You can reduce student resistance in three ways.
- First, prepare your students by telling them that the goal of these tasks is to get them thinking mathematically; you are looking for their ability to define and evaluate a measure. You may need to "sell" these tasks to the students since many will not be accustomed to open-ended problems that don't assess manipulation of formulas. You may choose to emphasize the ubiquity of measures in all sciences, and in everyday life, with examples of well-defined measures (chosen from science e.g., heat and temperature) and less well defined measures ('happiness'; 'quality of teaching'). You might flag the importance of good measures for any kind of modeling in mathematics and science.
- Second, to help your students adjust to these types of problems, the first "Creating Measures" task should be a non-graded, in-class, group-based exercise. For, if the task is non-graded, students can work on it without fear of "messing up their grade" and, if the task is done in-class, students can receive help from you as they work through the exercise. And, 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).
Providing guidance as students work on 'Creating Measures' tasks
Clear guidance is provided for each task in the downloaded materials. The amount of guidance that students need should decrease as they become familiar with these types of problems. The amount and type of help you provide the students depends upon your goals for the task. For instance, if your primary goal is to assess how well students have learned to create measures in the course of teaching, you may choose to provide very little assistance.
Reporting out of individual or group work
If you decide to have the students congregate as a large group to discuss their solutions, it is again helpful to decide the degree to which you will participate in these discussions (which will depend upon your goals for the session). For instance, you can facilitate the students' discussion, having them defend their ideas and write their ideas on the board, while adding almost none of your own. Your comments might be designed to encourage students to critique their own, and others', measures in a more sophisticated way. Or, you might lead the discussion, soliciting student measures and offering a structured critique of the advantages and disadvantages of different measures. Critiques using characteristics of good and poor measures (e.g., insensitivity to scale; appropriate rank order with respect to 'intuitive' orderings; or, measures that are bounded) will reinforce the key ideas in the CAT.
Formal and informal use of "Creating Measures" tasks
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. However, as students gain experiences with these kinds of tasks, the amount of coaching you do should decline and students should rely less on this kind of assistance.
Group work versus individual work
The open-ended nature of "Creating Measures" 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 web 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 "Creating Measures" tasks is that they require little mathematical technique. Students do need to have a basic knowledge of geometry concepts (area, perimeter, length), basic numeric skills (multiplication, addition, subtraction, division), use of formulas, and some algebra (the notion of a variable). The types of measures that students create will depend upon their mathematical knowledge, for the more sophisticated a student is with mathematical tools, the more likely the student will be able to come up with more sophisticated measures. However, the quality of a solution lies not in the complexity of the mathematical form, but in its functionality.
- Prepare by reading through the "Creating Measures" 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 'Creating Measures' task, emphasizing that they should be able to defend both their assumptions and the reasoning that leads to their answer, and the desirable 'metric' functions of the measure.
- 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