Think Aloud
“Clinking Glasses”
Learning Intention: To engage in mathematical problem solving and develop an understanding for
the purpose of problem solving.
Success Criteria: You will be able to understand how discussing a problem and sharing solutions
strategies supports understanding in problem solving situations.
Slide 1: What does it mean to engage in problem solving? Jot down your thoughts to these questions.
 What comes to mind when you hear the words “problem solving”?
 What is the purpose of problem solving in mathematics?
 What do your student think of when they hear those words?
 What strategies do you use to help them?
Slide 2: Why engage in problem solving?
Solving problems is not only the most important end of mathematics; it is also a means for
learning mathematics. (Beckman, 2008)
Good problems:
 Deepen our thinking about mathematics
 Help us recognize connections between topics in mathematics
 Force us to confront mathematical misconceptions
Working on “good problems” we learn mathematics better! Even a simple one-step or twostep story problem can be a “good problem” for some students.
Handout Clinking Glasses Problem
Slide 3. Setting the Stage – Task Clarification
 Independently read the problem
 Visualize the situation
 Restate the problem (not focusing on the answer)
 Connect to real-life situations
Part 1:
1. Put your pencils down.
2. Read the problem independently.
3. When finished, turn your paper over.
Part 2:
1. Turn to another person at your table.
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
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2. Retell the problem to each other…Do not tell the solution or try to solve the problem. Just
retell the story that’s on the paper.
Part 3:
1. Who can visualize the situation?
2. Who is willing to retell the story to the whole class? (remove the problem from the
overhead)
3. Gather several versions of the story. Retell it until the story is told clearly as it is written on
the paper. Make sure to listen for information that is added through the retellings.
4. When have you seen this happen in real-life?
Slide 4: Clarification of Concepts and Vocabulary in Context; Making Connections
 Clarify vocabulary specific to the mathematics concepts
 Clarify vocabulary related to the context of the problem
 Connect the mathematical ideas to previous work
1. Look over the problem.
2. Select a word from the story that you think is important to understand in order to solve this
problem. Share your word with your partner.
3. Once you have shared your word, describe to your partner why your word is important to the
context of the story. (clink, pair of people, toast, everyone else)
Then chart vocabulary words from various groups…connect the words to the mathematics of the
problem situation. For example:
a. pairs-2 people, so each two people get one clink”
b. clink-sound when two glasses meet. Only one sound but two glasses
c. toast-raise glass and tap with another class
d. everyone-else-all other people in the room
Slide 5: Where To Start
 Discuss various approaches for entry into the problem
 Share reasoning on approaches
1. How might you start working on the problem and why? (don’t let people start to solve it)
2. Once a few strategies have been discussed, allow work time first individually and then working as a
small group...
Ideas for strategies: Start with a simpler related problem, Act it out, Draw a diagram, Make a table,
Look for a pattern or a relationship, Be organized, Be systematic.
Slide 6: Thinking about the Solution Strategy and the Mathematics
 Redefine the question in the problem


Survey students for models of what the solution strategy might look like
Relate connections between the answer, the solution strategy, and the problem
1. How did you think about the question in the problem?
2. What models were used to help think about a solution strategy?
3. What connections can be made between the answer, the solution strategy, and the problem?
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University of Wisconsin-Milwaukee and Milwaukee Public Schools
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Share out answers or solution strategies. Post on chart paper around the room.
Slide 7. Debrief: Thinking About the “Think Aloud” Process...
 Think about the “think aloud” process used today to introduce you to the problem.
 Review the steps with a neighbor (pass out Think Aloud directions)
 How was this process similar or different than what you do with your students?
Slide 8. Poyla’s Four Steps to Problem Solving
First:
Understanding the Problem
Second: Devising A Plan
Third:
Carrying Out The Plan
Fourth: Looking Back
Slide 9: What is a Think Aloud?



Based on a reading strategy, a “Think Aloud” in reading models what kinds of thinking a
good reader uses to comprehend text.
The use of a “Think Aloud” for math was developed by a MPS Mathematics Specialist and
an MPS Literacy Specialist
Implemented differently in mathematics in MPS.
o The “think aloud” process develops independent thinking by students when working
in problem solving situations.
Slide 10: What does a Think Aloud Do?
Through questioning students are able to…
 visualize the situation posed
 develop important mathematical concepts and connections
 clarify vocabulary
 focus on what is needed for an answer and develop points of entry into using a strategy for
solving the problem
Homework Assignment:
Think Aloud: Use the “Think Aloud” strategy to engage your students in two different problemsolving situations over the next two weeks. Write a short (no more than 2 page) reflection on:
(a) How did the process influence the way you thought about the problem before you taught it?
(b) How did students respond to the process? Also comment on how the process worked with
students who struggle with problem solving situations, in what ways did it support their learning?
How did the process engage these struggling learners in more meaningful problem solving?
(c) Summarize some insights gained from the first time you tried the strategy as compared to the
second time.
(d) Make sure to include the problems you gave and some background on the students (e.g.,
grade, how many).
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
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Learning Intention
To engage in mathematical problem
solving and develop an understanding for
the purpose of problem solving
Success Criteria
You will be able to understand how
discussing a problem and sharing
solutions strategies supports
understanding in problem solving
situations.
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
4
What does it mean to engage in
problem solving? Jot down your
thoughts to these questions.
 What comes to mind when you hear
the words “problem solving”?
 What is the purpose of problem
solving in mathematics?
 What do your student think of when
they hear those words?
 What strategies do you use to help
them?
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
5
Why engage in problem solving?
Solving problems is not only the most
important end of mathematics; it is also a
means for learning mathematics.
(Beckmann, 2008)
Good problems:
 Deepen our thinking about mathematics.
 Help us recognize connections between
topics in mathematics.
 Force us to confront mathematical
misconceptions.
Working on “good problems” we learn
mathematics better!
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
6
Thinking About Clinking Glasses!
Setting the Stage
Part 1
 Put your pencils down.
 Read the problem independently.
 When finished, turn your paper over.
Part 2
 Turn to another person at your table.
 Retell the problem to each other.
 Do not tell how you would solve the
problem or try to solve the problem.
Just retell the story that’s on the
paper.
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
7
Clarifying Concepts
and Vocabulary in Context
Look over the problem.
 Select a word from the story that you
think is important to understand in
order to solve this problem. Share
your word with your partner.
 Once you have shared your word,
describe to your partner why your
word is important to the context of the
story.
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
8
Where To Start…
How might you start working on the
problem and why?
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
9
Thinking about the Solution Strategy
and the Mathematics

How did you think about the question in
the problem?

What models were used to help think
about a solution strategy?

What connections can be made
between the answer, the solution
strategy, and the problem?
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
10
Thinking About
the “Think Aloud” Process….
 Think about the process used to
introduce the problem.
 Review the steps with a neighbor.
 How was this process similar to what
you do with your students? Different?
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
11
Polya’s Four Steps to Problem Solving
First:
Understanding the Problem
Second: Devising A Plan
Third:
Carrying Out The Plan
Fourth:
Looking Back
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
12
What is a Think Aloud?
 Based on a reading strategy, a “Think
Aloud” models what kinds of thinking a
good reader uses to comprehend text.
 Think Aloud for math was developed
by a MPS Mathematics Specialist and
an MPS Literacy Specialist.
 Implemented differently for math, the
strategy develops independent
thinking by students when working in
problem solving situations.
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
13
What does a Think Aloud Do?
Through questioning, students
are able to…
 Visualize the situation posed.
 Develop important math concepts.
 Clarify vocabulary.
 Focus on what is needed for an
answer.
 Develop points of entry into using a
strategies for solving the problem.
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
14
Classroom Connections:
Think Aloud for Problem Solving
Use the “Think Aloud” strategy to engage your
students in two different problem-solving situations
during the next two weeks.
Write a short (no more than 2 page) reflection on:
(a) How did the process influence the way you
thought about the problem before you taught it?
(b) How did students respond to the process?
Also comment on how the process worked with
students who struggle with problem solving
situations, in what ways did it support their
learning? How did the process engage these
struggling learners in more meaningful problem
solving?
(c) Summarize some insights gained from the first
time you tried the strategy as compared to the
second time.
(d) Make sure to include the problems you gave
and some background on the students (e.g.,
grade, how many).
Math Alliance Project, January 2010
University of Wisconsin-Milwaukee and Milwaukee Public Schools
15