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Teaching School Mathematics through Problem Solving

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Recall facts and procedures from memory? Solve story problems? ... New York: Teachers College Press, 2000] A different sort of classroom. Grade 7 ... – PowerPoint PPT presentation

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Title: 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?

4
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)

5
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

6
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

7
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.

9
Another example The cylinders problem
  • LAUNCH Do cylinders with the same surface area
    have the same volume?

10
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.

11
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

12
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?

13
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.

14
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
15
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

16
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
17
Break into small groups
Teaching Math through Problem Solving
Nationellt Centrum för Matematikutbildning vid
Göteborgs Universitet
18
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.

19
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

20
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

21
Challenges to teaching through problem solving
  • Time
  • Need for support
  • Curriculum
  • Alignment with assessment practices
  • Unfamiliar role for (some) teachers

22
Benefits of teaching through problem solving
  • Promotes understanding
  • Is motivating
  • Helps memory
  • Enhances transfer
  • Influences attitudes and beliefs
  • Promotes the development of autonomous learners
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