Problem Solving across the

Curriculum

Ken Wolff, Mika Munakata and

Mary Lou West

Montclair State University

Montclair, New Jersey, USA

wolffk@mail.montclair.edu

munakatam@mail.montclair.edu

westm@mail.montclair.edu

• 243 acre campus on

a hilltop

• 23 km west of New

York City

• About 13,000

ethnically diverse

undergraduates and

4,000 graduate

students

• Undergraduates

educated in the

liberal arts tradition

• 2nd largest and fastest growing New

Jersey university

• Public university with 465 faculty

members

• Founded in 1908

• 250 undergraduate and masters’

majors, minors, concentrations, and

doctoral degrees

•Sc.D. in Audiology

•Ed.D. in Pedagogy

•Ed.D. in Mathematics Pedagogy

•Doctor of Environmental Management

MSU Mathematics Teacher

Education Program

• Preservice/undergraduate

• MS in mathematics, with a concentration in

mathematics education

• Middle School Certificate

• Middle School MA

Problem Solving=Solving Problems?

Constructivism

Piaget

Dewey

Polya

Schoenfeld

Vygotsky

Gardner

Constructivism

• Building on students’ experiences

• Learning takes place in a social context

• Actively engage students in their learning

Retention of Learning*

Students generally remember ..

10% of what they read

20% of what they hear

30% of what they see

50% of what they hear and see

70% of what they say and write

90% of what they say as they do something

* Adapted from Edgar Dale’s Cone of Experience

1

2

3

4

5

6

7

8

9

Coding for 1 through 9

For example, 4 8 is coded as

Please memorize each number and

its code

What is the code for 6 5 2 3?

1 2 3

4 5 6

7 8 9

1

2

3

4

5

6

7

8

9

1 2 3 4 5

6 7 8 9 10 11 12

13 14 15 16 17 18 19

20 21 22 23 24 25 26

27 28 29 30 31

13

Using the "Color the Board" game to

challenge students' notions of mathematics.

14

10

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9 10

Sample completed board

15

Rules:

• Four colors are used to color the 100 squares on a 10 x 10 board

• Each of the four colors makes a rectangle, for a total of four rectangles on the board

• Participants ask for (guess) the colors of squares to be revealed, one at a time

• Any square whose color can be determined without a guess is called a freebie.

16

Objective:

As a class, determine the location and color

of the four rectangles, minimizing guesses

10

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9 10

Is this mathematics?

• Common notions (misconceptions) about

the nature of mathematics

• What mathematical ideas and skills are

evident in this activity?

• What are some extensions of this activity?

Where is your blind spot?

Interdisciplinary connections that

interest middle school students

Purpose of this science activity

• Connecting math to science

• Connecting math to your own body

• Connecting math to social studies

• Using different learning styles:

– Visual

– Hands-on measurement

– Trigonometry, algebra, calculation

Measure the distance from your

eye to the paper when the star

first disappears

Calculate the ratio

• Ratio = separation of dots/distance to paper

• = (6 cm) / (about 19.5 cm)

• Angle1 = tan-1 (Ratio)

• = about 17 degrees

Measure the distance from your

eye to the paper when the star

first reappears

Calculate the new ratio

• Ratio = separation of dots/distance to paper

• = (6 cm) / (about 16.5 cm)

• Angle2 = tan-1 (Ratio)

• = about 20 degrees

Angular size of your blind spot

• = Angle2 – Angle1

• = about 20 degrees – 17 degrees

• = about 3 degrees

Actual width of your blind spot

• Diameter of a normal eye is 1.7 cm

• Width of blind spot = (1.7 cm) tan (angular

size)

• = (1.7cm) tan (about 3 degrees)

• =.09 cm = about 1 mm on your retina

“When are we ever going to need

this?”

• There are patterns in nature.

• The same patterns have the same solutions.

• This is the power of mathematics.

• The power of science is to recognize the

patterns.

Purpose of this science activity

• Connecting math to science

• Connecting math to philosophy

• Learning to use large numbers

• Using different learning styles:

– Visual

– Algebra

– Imagination

Expansion of the universe

Astronomers can measure the distance to

a galaxy.

They can also measure the velocity of

that galaxy away from the solar

system.

A graph of velocity vs. distance is a

straight line through (0, 0).

What does this graph mean?

Velocity = (distance) (a constant slope)

Velocity = distance / TIME

This TIME = the same for all galaxies.

= 13.7 billion years

This means that the universe is expanding

linearly since 13.7 billion years ago!

Where is the center of the

universe?

Note: a linear equation has an interesting

property:

It is always self-similar.

You can never determine where you are on a

line, unless there is a grid.

In the universe there is no grid.

106%

Discussion

10.6

KILOS

8.5

KILOS

6.1

KILOS ? KILOS

Chicken Weights

Paper Folding

Resources

• The Mathematics Teacher

• The Physics Teacher

• Teaching Mathematics in the Middle Grades

• Edgar Dale, Audiovisual Methods in Teaching, 1969

• John Dewey, Democracy in Education, 1916

• Howard Gardner, The Unschooled Mind: How Children Think and How Schools Should Teach, 1991

• D. Downie, T. Slesnick, and J.K. Stenmark, Math for Girls and Other Problem Solvers. EQUALS, 1981.

• Jean Piaget, To Understand Is To Invent, 1973

Resources (continued)

• George Polya, How to Solve It, 1945

• Project ASTRO Resource Notebook, Astronomical

Society of the Pacific

• Stephen K. Reed ,Word Problems: Research and

Curriculum Reform, 1999

• Alan Schoenfeld, Mathematical Problem Solving, 1985

• Lev. S Vygotsky, Mind in the Society, 1978

What is the sum of

all the digits of the numbers in

the sequence

1, 2, 3, 4, … , 97, 98, 99, 100?

A man is standing on a bridge as shown below.

500 m 300 m

A train is approaching from the right. If the man

runs towards the train at 10 km/hour he will get to

the end of the bridge at the same time as the train.

If he runs to the other end of the bridge at the

same rate he will also get there just as the train

does.

How fast is the train traveling?

Towers

• 3 pegs

• N discs on the right most peg

• Move all discs to the left most peg by

moving one disc at a time AND never

placing a larger disc on top of a smaller disc

• How many moves does it take to do this?