Problem Solving across the Curriculum Ken Wolff, Mika Munakata and Mary Lou West Montclair State University Montclair, New Jersey, USA [email protected] [email protected] [email protected]
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Problem Solving across the
Curriculum
Ken Wolff, Mika Munakata and
Mary Lou West
Montclair State University
Montclair, New Jersey, USA
• 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?
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