No Heads: The Juicy Middle of Real Word Problems
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
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
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.
• Equations and
• Headless Task Template
Tim McCaffrey is the founder of Agree
or Disagree?, writes at http://timsmccaffrey.com/, 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.