# Teaching Mathematics through Problem Solving - PowerPoint PPT Presentation

View by Category
Title:

## Teaching Mathematics through Problem Solving

Description:

### A problem-centered approach to teaching mathematics uses interesting and well ... Math Trailblazers (www.math.uic ... SPEAKING POINTS Elementary ... – PowerPoint PPT presentation

Number of Views:2542
Avg rating:3.0/5.0
Slides: 60
Provided by: Messe8
Category:
Tags:
Transcript and Presenter's Notes

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
Problem Goldenberg 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
Problem Goldenberg 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
• ? Does this remind you of any other problems you
have solved?

12
Teaching Mathematics through Problem Solving
Research Perspectives M. 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 Perspectives M. 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 Perspectives M. 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
• ? 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
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
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
52
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)
53
54
Web Resources
54
55
Web Resources
55
56
Web Resources
56
57
Web Resources
57
58
Web Resources
58
59