Teaching Mathematics through Problem Solving

Emma Ames Jim Fey Mary Jo Messenger Hal

Schoen

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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)

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Selecting Classroom Tasks - Basic Questions

R. Marcus J. T. Fey

- Will working on the tasks foster students

understanding of important mathematical ideas and

techniques?

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?

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?

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?

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?

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?

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?

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?

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.

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.

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?

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?

What Happens in the Classroom When Mathematics

is Taught Through Problem Solving?

In addition to learning mathematics, students

learn to be good problem solvers.

What Happens in the Classroom When Mathematics is

Taught Through Problem Solving?

- Thinking and problem solving are the fundamental

part of our lessons.

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

Team Work

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?

Problems must have meaning for students.

Teaching Equation Solving and Inequalities

Through Problem Solving

- Cable TV (CPMP Year 1)

5 2.5X 75 2.5X

Teaching Equation Solving and Inequalities

Through Problem Solving

- Cable TV (CPMP Year 1)

30 5 2.5X

Teaching Equation Solving and Inequalities

Through Problem Solving

- Cable TV (CPMP Year 1)

75 2.5X gt 40

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.

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

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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?

Teaching Equation Solving and Inequalities

Through Problem Solving

- Lines (CPMP Year 1)

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?

Teaching Equation Solving and Inequalities

Through Problem Solving

- Lines (CPMP Year 1)

Teaching Equation Solving and Inequalities

Through Problem Solving

- Lines (CPMP Year 1)

The Bears Problem

The Bears Problem

- Various Levels
- Middle School
- Algebra
- Precalculus

The Bears Problem

The Bears Problem

The Bears Problem

The Bears Problem

The Bears Problem

The Bears Problem

The Bears Problem

The Bears Problem

(No Transcript)

The Bears Problem

Learning Through Problem Solving

- Students Actively Participate, Reason, and

Explain to Others

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.

Time to Reflect

Frustration is Part of a Real Problem

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.

Teaching Through Problem Solving

- Always be aware of who is doing the thinking, the

teacher or the student.

Byproducts

- Self esteem
- Motivation
- Better Understanding

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)

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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

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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