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Interpreting Statistics: A Case of Muddying the Waters
During the
  • Review Individual Solutions to Muddying the Waters (10 minutes)
      • Remind students of their work on the assessment task.
        • Recall the work you did in the last lesson on river pollution.
        • In this lesson, you will build on that work.
      • Return the papers to the students.
      • If you chose to write questions on the board rather than on individual papers, display them now.
        • I read your papers, and I have some questions about them.
        • I'd like you to work on improving your answers for a few minutes, using my questions.
      • Ask students to work on their own for a few minutes, answering your questions.
  • Interactive Role-Play Introduction (10 minutes)
      • To introduce the lesson task, use a projector and slides P-3 to P-9 from the projector resource: Exhibits 1, 2, 3, 4; Scene 1: The assistant DA's office; Scene 2: At the Factory.
      • If you do not have a projector, hand out the printed copies of these slides.
      • You also need four copies of the role-play script, A Case of Muddying the Waters (one copy for each actor and one copy for yourself).
        • In the next section of this lesson, you will be working on river pollution again.
        • The role-play sets the scene.
      • Ask the actors to read out the script. Advise them to talk slowly, and to pause at the end of each sentence as the script contains a lot of information.
      • Encourage the students to listen carefully to the facts being presented about the river pollution.
  • Collaborative Small-Group Work (25 minutes)
      • Once the students have acted out the scenes (and the applause has died down) turn to the class and say:
        • The case goes to court. The Assistant DA prosecutes the Factory Owner for polluting the river.
        • What does "prosecution" mean?
        • Your task now is to be the judges. You have to reach a fair judgment about who wins the case.
      • Organize students into groups of two or three.
      • Provide each small group with a copy of the worksheet, Case Notes.

      Case Notes contains the map, information from the script, and arguments made in court by the Environmental Officer and the Factory Owner.

      • Help students to understand the task and its context.
        • I'm giving you a copy of the arguments presented in court.
        • Read through the information carefully. Write notes on what you think the data and statistics show.
      • In particular, ask them to focus on critical analysis of the information presented.

        • Explain why you agree or disagree with the arguments people made using math.
        • The important thing is to look critically at all of the information. Do not just accept what people say as fact.
        • At the end of the lesson, you will use your work to decide together whether the factory owner is guilty or not guilty of polluting the river.

      These instructions are reproduced on slide P-8, Judge's Instructions.

      During small-group work you have two tasks, to notice strengths and weaknesses you see in students' work, and to support their thinking.

      Note strengths and weaknesses in students' work.

      Find out about students' current levels of understanding and the difficulties they may encounter in the task. Students may be used to interpreting statistical diagrams, but may find it more difficult to critique someone else's biased reading of information. Students may fail to notice a bias in a question or may struggle to understand the issue of small sample size. You can use the information about common difficulties to focus the whole-class discussion towards the end of the lesson.

      Support student thinking.

      Try not to solve students' difficulties for them. Instead, ask them questions to help them move their thinking on.

        • You could strengthen your argument if you did some math on the data you've been given.
        • Is there another way to present this data?
        • Could you redraw that chart so it displays the important features of the data better?

      Questions similar to those in the Common Issues table were found to be useful in lesson trials.

      • For students who are struggling, it may help to ask some specific questions about aspects of the mathematics:
        • Describe this chart.
        • Is there another way to present this data?
        • The Environmental Officer/Factory Owner drew this conclusion. Can you draw any different conclusions from this evidence?
      • Encourage students to explain their reasoning to others in the group before writing it down. Other group members may question and refine the explanations.
  • Whole-Class Discussion: Reaching a Judgment (10 minutes)
      • Organize a whole-class discussion, focusing on the mathematical practice of critiquing the reasoning of others.
      • Choose a group to present their argument about one piece of evidence.

      Instruct the other groups to listen and write down questions about the group's argument.

        • Hani, does this evidence help the factory owner show he is not guilty? Tell us why you think that.
        • If you disagree with the group's interpretation of the evidence, write why, and challenge them at the end of their presentation.
      • Once the group has presented their case, other groups get a chance to challenge the details of their argument.
      • If the challenge is not based on mathematics, you can rule it out of court.

        • That is not a mathematical argument. As there is no good evidence, it can't be accepted in court.
      • Once you have modeled this process for students, give them responsibility for deciding whether there is an evidential base for each claim and challenge.
        • Is that a mathematical argument? Is there good evidence for what [Shelley] has just said?
        • Does the evidence support her conclusion?
      • Once students have worked through the four pieces of evidence, ask them to come to a collective decision about the verdict.
        • Do you find the factory owner guilty or not guilty?

      If there is disagreement, take arguments from both sides. You may find you cannot reach a collective decision. In that case, suggest students send the defendant for a retrial.

  • Whole-Class Discussion: Summing Up (5 minutes)
      • Point out that an important message of this lesson is that it is easy to "get it wrong" when interpreting statistics, especially in complicated real-world situations.
      • In reality, most of the "evidence" in this lesson is too vague to draw any firm scientific conclusions. A lot of questions are left unanswered.

        • How exactly were the wildlife surveys conducted? Why did the second survey look at so many more fish than the first? How do you "count the number of invertebrates" at a site?
        • The dam has reduced the flow of the river by 80%. Even without the pollution, is it possible that this could affect the wildlife or the popularity of the Riverside Center?
        • Can you see any other problems with the data collection? With the statistics that have been calculated?
      • If you think it is appropriate to the class, you could mention that if they study statistics further, they will learn how to calculate significance: the likelihood that a difference in two results is not just "the luck of the draw."

      Most serious scientific studies will do this, but you do not often find it in news reports!

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