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Interpreting Statistics: A Case of Muddying the Waters
Before the
  • Interactive Whole-Group Introduction (10 minutes)
    • Have the students do this task in class a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will then be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of the task Muddying the Waters.
      • Ask students to read through the task carefully.

      Some teachers ask students to take turns to read parts of the task aloud.

      • Introduce the task, asking questions to help students understand the problem and its context.
        • This task is concerned with the river pollution and its effect on the environment. What do I mean by river pollution?
        • What does it mean when someone says that the level of pollution in a river is illegal?
        • Does anybody know of a river that is polluted? What was the source of the pollution?
        • How can you tell that a river is polluted?
      • In particular, explain how chemical pollution in a river is measured. You could use a teaspoon to help illustrate this.
        • Chemical pollution is measured in milligrams per cubic meter of river water.
        • Does anyone know how much a milligram is? A cubic meter is?
        • A teaspoon of sugar is about 4,000 mg. This classroom is about 300 cubic meters. [Use figures for your own room.]
  • Assessment Task: Muddying the Waters (15 minutes)
      • Ask students to work through the task.
        • Spend fifteen minutes working individually on this task.
        • Don't worry if you can't complete everything.
        • There will be a lesson [tomorrow] that should help you understand the math better.
        • Your goal is to be able to answer questions like these confidently by the end of the next lesson.
      • At this stage, we suggest that you do not help students with the task or comment on their work. Stop them after 15 minutes, whether or not they have finished.
      • Collect students' papers for analysis.
  • Preparation for the Next Lesson (5 minutes)
      • Explain the theme for the next lesson.
      • You need three copies of the role-play script, A Case of Muddying the Waters

        • Next lesson we are going to continue the theme of river pollution with some role-play.
        • The role-play sets the scene for the task. An Assistant DA talks to a local Environmental officer and a Factory owner about the pollution of a river. Who wants to play these three roles?
      • Give each volunteer a copy of the role-play script, A Case of Muddying the Waters.
      • Ask the volunteers to read over their parts before the next lesson.

      If you wish, enrich the role-play by asking the volunteers to bring in a few theatrical props.

  • Assessing Students' Responses
    • Collect students' responses to the task. Make some notes on what their work reveals about their current levels of understanding. The purpose of this is to forewarn you of issues that will arise during the lesson itself, so that you may prepare carefully.

      We suggest that you do not score students' work. The research shows that this will be counterproductive as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics.

      Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given in the Common issues table (below). These have been drawn from common difficulties observed in trials of this lesson unit.

      We suggest that you write a list of your own questions, based on your students' work, using the ideas below. You may choose to write questions on each student's work. If you do not have time to do this, select a few questions that will be of help to the majority of students. These can be written on the board at the beginning of the lesson.

      Common Issues: Suggested Questions and Prompts:

      Student does not understand that there are alternative interpretations of data and statistics, some of which may be biased.

      • For example: The student copies or paraphrases the Riverside Manager's interpretations.
      • What does the chart show? What does the Riverside Manager say the chart shows? Is there a difference?
      • In what ways is what the Riverside Manager said misleading?

      Student does not recognize that things can happen together without one causing the other.

      • For example: The student does not contradict the causal claim made about the scatter plot.
      • If two things happen at the same time, does that mean one made the other happen?
      • Write down another reason that there might be a correlation.
      • What other interpretations of the correlation can you find?

      Student does not understand that survey questions may push respondents towards a particular response.

      • For example: The student does not recognize that the phrasing of the statement biases respondents towards thinking of the river as polluted and smelly.
      • Does the way this question is asked make a "yes" response more likely than "no" or "maybe"? Why do you think that? Does it matter?
      • Try writing this question in a way that doesn't push the respondent towards a particular answer.

      Student does not recognize that statistics may be compiled in ways that push readers toward a biased interpretation.

      • For example: The student does not recognize that on the scatter plot, starting the "number of visitors" scale at 122 (rather than 0) distorts perceptions of the proportional change in the number of visitors.
      • Or: The student does not recognize that it is inappropriate to draw conclusions about the whole population from such a small sample size.
      • Notice that the scale on this graph starts at 122. How different would the graph look if you drew the axis showing the whole range? How might that affect your interpretation?
      • How many visitors were there overall? When was the survey conducted? Can you think of a more convincing way to set the survey?
      • How do the ways data was collected affect your interpretation of the results?
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