No Heads: The Juicy Middle of Real Word Problems

• # No Heads: The Juicy Middle of Real Word Problems

By Tim McCaffrey, posted May 23, 2016 –

Last post, we discussed the concept of presenting Tailless word problems by chopping off the question and allowing student to make sense of the information and to formulate their own question(s). In this post, I am going to discuss Headless tasks.

To create a Headless task, simply chop off the information and just give the students a question. You may need to give some (least amount possible) information, so that there is some context to the problem.

Step 1: Remove the information and present a question.

Isn’t this how mathematics really works in the real world? We start with a question that we wonder about, and we have to go out and search for relevant information. Below I have taken a problem and removed all the information:

You and two of your friends saved money for four weeks. Which of you saved the greatest and least amount of money each week?

The goal is to be less helpful and give the least amount of information to allow students to think about the information they need to solve the problem. The only information I give is the first statement. This leaves room for much wondering.

Step 2: Students write and share information needed to solve the problem.

Whatever information students ask for, inquire about their reasoning. Ask them what they would do with the information they are asking for. Would it be enough information? Have them share with their classmates to see if they agree or disagree. This step continues to remain low floor because no calculations are required and students can make sense of the information they need to solve the problem.

Step 3: Mathematize the information.

My favorite part of a headless task is mathematizing the information that students want. For instance, students may ask for the amount of money everyone saved in the first week. Then you give them the information shown below.

I have done two things here:

1. Mathematized the information, by giving each savings in different representations; and

2. Withheld information.

After giving students this information, let them debate if it is enough to determine the total savings of each person. I love what Dan Meyer shares, “Once you give information you cannot take it back.” Purposely withhold information and allow students to make sense of what they do have, and they will start asking better questions.

In review:

• Present a question with “less helpful” information.
• Allow students to determine what information is needed.
• Ask students how the information would help them with the problem.
• Give students information in part and mathematize the information.