Teaching School Mathematics through Problem Solving

1
Teaching School Mathematics through Problem
Solving

• Diana V. Lambdin
• (lambdin_at_indiana.edu)
• Frank K. Lester, Jr.
• (lester_at_indiana.edu)
• Indiana University

2
The world is different!

• Information age
• Global economies
• New skills and concepts are needed
• The mathematics curriculum must prepare students
  for this new world.
• This means that what we teach and how we teach
  must change!

3
What does it mean to understand mathematics?

• Do well on national and international
  assessments?
• Recall facts and procedures from memory?
• Solve story problems?
• Apply what is learned in school in everyday
  situations?
• IS THERE SOMETHING ELSE?

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Students understand a mathematical concept or
skill when they

• Demonstrate reasonable proficiency in using it.
  (proficiency)
• Use it to solve problems in a variety of contexts
  (and not only in school) (application)
• Know how it relates to other mathematical ideas
  (connection)

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A classroom scenario

• Grade 7
• Measurement unit on area
• perimeter
• Teacher goes over last nights
• homework
• Teacher reviews definitions
• formulas
• Teacher shows how to find area
• perimeter

• Students begin work on a set of 20 problems

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Homework

• Complete work on the 20 problems
• Solve the following word problem
• A shop owner wants to put a new wood floor in
  his shop. The shop is 12 meters long and 8 meters
  wide. How many square meters of floor will the
  wood cover?
• Adapted from Stein et al., Implementing
  Standards-based Mathematics Instruction. New
  York Teachers College Press, 2000

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A different sort of classroom

• Grade 7
• Measurement unit on area perimeter
• Students begin work on a challenging task in
  small groups
• Work continues for the entire class period using
  whatever tools the students need
• Teacher goes from group to group, asks questions,
  gives hints, but does not give answers

8
The rabbit pen problem

• Ms. Gomezs class will raise rabbits for their
  spring science fair. They have 24 meters of
  fencing to build a rectangular rabbit pen for the
  rabbits.
• 1. If Ms. Gomezs students want their rabbits
  to have as much room as possible, how long would
  each side of the pen be?
• How long would each side be if they had only 16
  meters of fencing?
• 3. How would you determine the pen with the
  most room for any amount of fencing? Organize
  your work so that someone else who reads it will
  understand it.

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Another example The cylinders problem

• LAUNCH Do cylinders with the same surface area
  have the same volume?

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EXPLORE
Questions for students

• Will all the cylinders hold the same amount?
  Explain your reasoning.
• How does changing the height affect the
  circumference? Explain.
• How does changing the circumference affect the
  volume? Explain.

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SUMMARIZE

• Have students report about their findings.
• Encourage student-to-student questions.
• Look back How is this problem related to
  problems we have done before?
• What have we learned about the relationship
  between circumference and volume?
• Examine the formulas for surface area and volume
  (Big math ideas)
• SA (2p)RH V pR2H

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Extending the Activity

• Have students conjecture about what is happening
  to the volume as the cylinder continues to be
  cut, getting shorter and shorter (and wider and
  wider).
• Some students may become interested in exploring
  the limit of the process of continuing to cut the
  cylinders in half and forming new ones.
• What if the cylinders have a top and bottom?

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Qualities of the Lesson

• A question is posed about an important
  mathematics concept.
• Students make conjectures about the problem.
• Students investigate and use mathematics to make
  sense of the problem.
• The teacher guides the investigation through
  questions, discussions, and instruction.
• Students expect to make sense of the problem.
• Students apply their understanding to another
  problem or task involving these concepts.

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The Teachers Primary Role in TMTPS
Teaching Math through Problem Solving

• Coach, facilitator, designer, listener and
  observer
• BUT NOT
• Lecturer, expert (source of the best way to do
  math)

Nationellt Centrum för Matematikutbildning vid
Göteborgs Universitet
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Teaching Actions

1. Provide hints as needed (focus or direction
   hints)
2. Give early solvers a challenge
3. Require all students to answer the question
4. Discuss solution strategies with entire class
   (Have students evaluate their own work)
5. Look back and look ahead (What have we learned?
   Can we generalize? . . .)

• Motivate interest in the problem
• Read the problem with the students and discuss
  what the information tells you
• Ask Is this problem like others you have
  solved?
• (Optional) Discuss possible strategies
• Observe and question students to determine what
  they are thinking

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Parts of a problem-solving lesson
Teaching Math through Problem Solving

• Launch (Actions 1 - 4)
• Explore (Actions 5 - 8)
• Summarize (Actions 9 - 10)

Nationellt Centrum för Matematikutbildning vid
Göteborgs Universitet
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Break into small groups
Teaching Math through Problem Solving
Nationellt Centrum för Matematikutbildning vid
Göteborgs Universitet
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Beliefs about Teaching through Problem Solving

• Problem solving can serve as a vehicle for
  learning new mathematical ideas and skills.
• A problem-centered approach to teaching
  mathematics uses interesting and well-selected
  problems to launch mathematical lessons and
  engage students.
• Good problems can inspire exploration of
  important math, nurture persistence, reinforce
  the need to understand strategies, math
  properties and relationships.

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And also --

• PS ability develops over time and through
  prolonged involvement
• Math concepts, processes, and procedures interact
  and develop concurrently
• Instruction must be systematic, organized, and
  teacher has a different role
• Content is organized around big ideas
• Lessons often involve group work, reflective
  writing, alternative assessment

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New roles and new expectations

• New roles for teachers
• New expectations for students
• Different kinds of tasks
• New emphases during instruction
• New types of assessment

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Challenges to teaching through problem solving

• Time
• Need for support
• Curriculum
• Alignment with assessment practices
• Unfamiliar role for (some) teachers

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Benefits of teaching through problem solving

• Promotes understanding
• Is motivating
• Helps memory
• Enhances transfer
• Influences attitudes and beliefs
• Promotes the development of autonomous learners