No Heads or Tails: Promote Students’ Sense Making and Perseverance

  • No Heads or Tails: Promote Students’ Sense Making and Perseverance

    By Tim McCaffrey, posted May 9, 2016 –

    Think of the last word problem you gave your students. Did any of your students—

    • read it and decide to skip it?
    • read the first line and decide to skip it?
    • skip it all together?
    • dispense with reading it and just guess the answer?
    • pick numbers and apply an operation?
    • ask for help (AKA wanting the solution steps)?
    • ask for help (AKA wanting the answer)?

    If so, then keep reading because this blog post is for you and your students. The following steps are the secret to unlocking students sense making and perseverance in solving real-word problems:

    Step 1: Remove the question and ask students to write down what they know for sure.

    Two large storage tanks, T and W, contain water. T starts losing water at the same time additional water starts flowing into W. The graph below shows the amount of water in each tank over a period of time. Assume that the rates of water loss and water gain continue as shown.

    2016-05-09 art1

    Removing the questions and asking student to write down what they know for sure allows all students to access the task. It also allows students to focus on making sense of the information rather than on the question.

    Allow students to convince one another of the information they know for sure. As you travel from group to group, you will learn what students know and possibly some of their misconceptions as well.

    Step 2: Ask students to write down question(s) that make sense to ask.

    Given time to make sense of the information and discussing it with peers, students can now write down questions that make sense to them. Have students share their question(s) in groups and possibly vote on the most interesting one. Record the students’ questions and the question(s) that you want your students to address. Here is the kicker, “You can choose any question we came up with as a class. However, you also have to do my question as well.” This will allow buy-in and give students ownership of the question of their choice as well as the teacher question.

    For our prompt above, here is my question: In three different ways, find when the amount of water in the two tanks is the same.

    Step 3: Collect data as students work on their problems.

    This is probably one of the most important steps because these data will be used to make your next move. In other words, depending on how students are responding to the task, you may need to—

    • do some instruction;
    • have groups share their strategy with the class;
    • address individual groups questions;
    • pose purposeful questions to advance students thinking; and/or
    • sequence the order that groups share their work.

    Step 4: For those students who finish early, give an extension question.

    We all have those students who need an additional push. Having a few extension questions in your back pocket is valuable to keep all students engaged and thinking. Here would be my simple extension: Assume that the rates of water loss and water gain are not constant. How does that change your responses?

    This is called a Tailless word problem because the tail (question) has been cut off. It’s amazing how much a task can be opened up by making such a simple adjustment. The next time you pose a word problem to your students, chop off the question and unlock sense making for your students. Next post, we will discover how Headless problems reinforce sense making and perseverance.

    In review:

    • Present a task with no question(s).
    • Ask students to write what they know for sure.
    • Allow students to share what they know and prove it to one another.
    • Ask students to write down questions that make sense to ask, given the information.
    • Extension: Ask a question that requires more information, and wait for the students to ask for it.


    2016-05 McCaffrey aupic Tim McCaffrey is the founder of Agree or Disagree?, writes at, and tweets at @timsmccaffrey. He currently serves as the mathematics coordinator for grades 6–12 in Fontana, California. He desires to help coaches and administrators implement sound mathematical practices that will help students become deep thinkers.


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