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

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Classroom Assessment Techniques
'Reasoning from Evidence' Tasks

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Teaching Goals

Suggestions for Use
Introducing 'Reasoning from Data' tasks for the first time.
Many students will find the tasks unfamiliar. In statistical work, students are usually presented with "clean" data (e.g., data already aggregated into tables) and are told which methods to use for its interpretation. In situations where students have conducted experimental work to gather data, often their teacher has told them which representations or methods to use. In our experience, students rarely have the opportunity to make such decisions for themselves.

Thus, the first time they see the "Reasoning from Data" tasks, it is helpful for them to spend some time discussing possible approaches to the task in pairs or small groups. They may even be able to share out some of the workload of aggregating results. Sometimes, one member of a group will suggest drawing a bar chart, while another will suggest a line graph or a table. They should be encouraged to do all of these and compare the relative advantages of each representation.

Occasionally, we find that students decide to use software (such as a spreadsheet program like Microsoft Excel) to analyze the data. This is not always as straightforward as it sounds, since it adds some complexity. Students will need to think about the format of the data as it is entered (e.g., decimals, times and dates, text) and choose sensible graphical representations to use when analyzing the output. It is easy to select meaningless graphical outputs!

Data are presented here both in printed form and as Excel spreadsheets. Your choice of how to present the data to students will be determined by your teaching goals.

Providing guidance as students work on 'Reasoning from data' tasks.
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 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 that the students struggle with solving the problems on their own (and learn that they can "do math"), you may choose to provide very little assistance; however, if your goal is that the students can understand and evaluate their reasoning process, then you may provide them with more assistance in that direction.

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, 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. This approach can direct students away from viewing you as the authority of the information. Or, you can lead 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 of 'Reasoning from data' 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 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.

Group work versus individual work
The open-ended nature of 'Reasoning from Data' 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 'Reasoning from Data' tasks is that they require very little mathematical knowledge, yet they allow students to use advanced mathematical knowledge where they do possess it.

Students do need to have a basic understanding of chance and proportional reasoning. They will also need to use a calculator and draw graphs when analyzing the data. Of course, as the choice of analysis tools are left to the student, they may decide to use more sophisticated methods, such as graphing facilities in spreadsheets, tests of significance and so on. However, the value of these tasks lies not in the sophistication of the methods used, but with students' ability to draw sensible conclusions from the data.

Step-by-Step Instructions

  1. Prepare by reading through the 'Reasoning from Data' 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 the your goals for the 'Reasoning from Data' task, emphasizing that they should be able to defend both their choice of method 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.

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

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