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Teaching Mathematics through Problem Solving

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Title: Teaching Mathematics through Problem Solving


1

Teaching Mathematics through Problem Solving
Emma Ames Jim Fey Mary Jo Messenger Hal
Schoen
1
2
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. In this way, new ideas,
    techniques, and mathematical relationships emerge
    and become the focus of discussion. Good problems
    can inspire the exploration of important
    mathematical ideas, nurture persistence, and
    reinforce the need to understand and use various
    strategies, mathematical properties, and
    relationships.
  • (Principles and Standards for School Mathematics,
    National Council of Teachers of Mathematics 2000,
    p. 182)

3

3
4
Selecting Classroom Tasks - Basic Questions
R. Marcus J. T. Fey
  • Will working on the tasks foster students
    understanding of important mathematical ideas and
    techniques?

5
Selecting Classroom Tasks - Basic Questions
R. Marcus J. T. Fey
  • Will the selected tasks be engaging and
    problematic, yet accessible, for many students in
    the target classes?

6
Selecting Classroom Tasks - Basic Questions
R. Marcus J. T. Fey
  • Will work on the tasks help students develop
    their mathematical thinkingtheir ability and
    disposition to explore, to conjecture, to prove,
    to represent, and to communicate their
    understanding?

7
Selecting Classroom Tasks - Basic Questions
R. Marcus J. T. Fey
  • Will the collection of tasks in a curriculum
    build coherent understanding and connections
    among important mathematical topics?

8
Interesting Variations on a Basic
ProblemGoldenberg Walter
  • Find the mean of 7, 4, 7, 6, 3, 8, and 7.
  • What if only five of the seven data are given?Can
    we determine the missing data if we know the mean
    of the original seven?
  • What if we compute the mean of each possible
    combination of only five of the given seven
    numbers? (How many such combinations are
    possible?) What could we learn from, say, a
    histogram of those means?

9
Interesting Variations on a Basic
ProblemGoldenberg Walter
  • Find the mean of 7, 4, 7, 6, 3, 8, and 7.
  • What if the original seven numbers are sampled
    from a population consisting of eight numbers?
    What might we reasonably infer about the eighth
    number? Do ideas from problem 2 help answer that
    question?
  • What if we know the mean but none of the data?
    What, if anything, could we say about the data?
    What possible sets of data would fit?

10
Some Questions That Promote Understanding - D. A.
Grouws
  • ? How did you decide on a solution method to
    try?
  • ? How did you solve the problem?
  • ? Did anyone solve it in a different way?
  • ? How would you compare these solution methods?

11
Some Questions That Promote Understanding - D. A.
Grouws
  • ? Which of the solution methods do you like
    best? Why?
  • ? Can you tell me how you solved the problem
    without saying the answer?
  • ? Does this remind you of any other problems you
    have solved?

12
Teaching Mathematics through Problem Solving
Research PerspectivesM. K. Stein, J. Boaler,
E. A. Silver
  • The research on TMPTS and on curricula designed
    to support it suggests both the feasibility and
    efficacy of this approach.
  • When TMPTS is implemented effectively, students
    (compared to those taught traditionally) are
    likely to better understand mathematical
    concepts, to be willing to tackle challenging
    problems, and to see themselves as capable of
    learning mathematics.

13
Teaching Mathematics through Problem Solving
Research PerspectivesM. K. Stein, J. Boaler,
E. A. Silver
  • TMPTS is challenging and to do it well teachers
    need support, including good curriculum materials
    and strong professional development.
  • TMPTS can work with a wide range of students, but
    the level of student support required may differ
    depending on the students mathematical
    background and interest.

14
Teaching Mathematics through Problem Solving
Research PerspectivesM. K. Stein, J. Boaler,
E. A. Silver
  • ? Which of the solution methods do you like
    best? Why?
  • ? Can you tell me how you solved the problem
    without saying the answer?
  • ? Does this remind you of any other problems you
    have solved?

15
Some Questions That Promote Understanding - D. A.
Grouws
  • How can we change the problem to get another
    interesting problem?
  • ? What mistakes do you think some students might
    make in solving this problem?

16
What Happens in the Classroom When Mathematics
is Taught Through Problem Solving?
In addition to learning mathematics, students
learn to be good problem solvers.
17
What Happens in the Classroom When Mathematics is
Taught Through Problem Solving?
  • Thinking and problem solving are the fundamental
    part of our lessons.

18
What Happens in the Classroom When Mathematics is
Taught Through Problem Solving? Technical
reading, writing, and communicating are
emphasized.
Just look at at this work young man.
Just look at this work young man.
Youve got some explaining to do.
Einstein as a boy
19
Team Work
20
What Happens in the Classroom When Mathematics is
Taught Through Problem Solving?
  • Real-world problems are used frequently and
    answers are given in terms of what makes sense
    for any given situation.
  • What is a Problem?

21
Problems must have meaning for students.
22
Teaching Equation Solving and Inequalities
Through Problem Solving
  • Cable TV (CPMP Year 1)

5 2.5X 75 2.5X
23
Teaching Equation Solving and Inequalities
Through Problem Solving
  • Cable TV (CPMP Year 1)

30 5 2.5X
24
Teaching Equation Solving and Inequalities
Through Problem Solving
  • Cable TV (CPMP Year 1)

75 2.5X gt 40
25
Teaching Equation Solving and Inequalities
Through Problem Solving
  • Cable TV

One way to solve the equations or inequality is
to make tables and graphs of (time, share) data
for the two models and look for key points in
each.
26

Tables and Graphs 30 5
2.5X 5 2.5X 75 2.5X
X Y1 Y2
0 75 5
1 72.5 7.5
2 70 10
3 67.5 12.5
4 65 15
5 62.5 17.5
6 60 20
7 57.5 22.5
8 55 25
9 52.5 27.5
10 50 30
11 47.5 32.5
12 45 35
13 42.5 37.5
14 40 40
15 37.5 42.5

26
27
Teaching Equation Solving and Inequalities
Through Problem Solving
Lines (CPMP Year 1) The next diagram shows linear
models from four rubber band experiments, all
plotted on the same grid. What does the pattern
of those graphs suggest about the similarities
and differences in the experiments?
28
Teaching Equation Solving and Inequalities
Through Problem Solving
  • Lines (CPMP Year 1)

29
Teaching Equation Solving and Inequalities
Through Problem Solving
Lines (CPMP Year 1)
  • (a). Sharing the work among your group members,
    make four tables of (weight, length) pairs, one
    table for each linear model, for weights from
    0 to 10 ounces.
  • (b).According to the tables, how long were the
    different rubber bands without any weight
    attached? How is that information shown on the
    graphs?
  • (c).Looking at data in the tables, estimate the
    rates of change in length for the four rubber
    bands as weight is added. How are those patterns
    shown on the graphs?

30
Teaching Equation Solving and Inequalities
Through Problem Solving
  • Lines (CPMP Year 1)

31
Teaching Equation Solving and Inequalities
Through Problem Solving
  • Lines (CPMP Year 1)

32
The Bears Problem
33
The Bears Problem
  • Various Levels
  • Middle School
  • Algebra
  • Precalculus

34
The Bears Problem
35
The Bears Problem
36
The Bears Problem
37
The Bears Problem
38
The Bears Problem
39
The Bears Problem
40
The Bears Problem
41
The Bears Problem
42
(No Transcript)
43
The Bears Problem
44
Learning Through Problem Solving
  • Students Actively Participate, Reason, and
    Explain to Others

45
Teaching Through Problem Solving
  • Establish the norms that students responses
    should include a rationale, students should
    strive to make sense of their own methods and
    those of their classmates, and students should
    ask questions and raise challenges when they do
    not understand.

46
Time to Reflect
47
Frustration is Part of a Real Problem
48
The Satisfaction of Solving the Problem
Well, then when you get your grades up to a B
average, THEN you can choose your own wallpaper.
49
Teaching Through Problem Solving
  • Always be aware of who is doing the thinking, the
    teacher or the student.

50
Byproducts
  • Self esteem
  • Motivation
  • Better Understanding

51
Materials to Support Teaching Mathematics Through
Problem Solving
Projects at All Levels The K 12 Mathematics
Curriculum Center (www.edc.org/mcc) Element
ary Projects The ARC Center
(www.arccenter.comap.com) Everyday Mathematics
(http//everydaymath.uchicago.edu)
Investigations in Number Data, and Space
TERC (www.terc.edu/investigations) Math
Trailblazers (www.math.uic.edu/IMSE/timsmath.htm
l)
51
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Materials to Support Teaching Mathematics Through
Problem Solving
Middle School Projects The ShowMe Center
(www.showmecenter.Missouri.edu/) Connected
Mathematics Project (www.math.msu.edu/cmp)
Mathematics in Context (www.ebmic.com)
MathScape Curriculum Center (www.edc.org/mathscape
) MATHThematics Project
(www.mcdougallittell.com/bookspots/math_thematics.
cfm) Pathways/MMAP Curriculum
(www.mmap.wested.org
52
53
Materials to Support Teaching Mathematics Through
Problem Solving
High School Projects COMPASS (www.ithaca.edu/com
pass) Core-Plus Mathematics Project
(www.wmich.edu.cpmp) Interactive Mathematics
Project (www.mathimp.org) MATH Connections
(www.mathconnections.com) Applications /Reform
in Secondary Education (www.comap.com/highschoo
l/projects) SIMMS Integrated Mathematics
(www.montana.edu/wwwsimms/Materials20.htm)
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Web Resources
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Web Resources
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Web Resources
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Web Resources
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Web Resources
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Your Challenge!
Education is not the filling of a pail, but the
lighting of a fire. William Butler Yeats
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