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# Lance Burger Math 149 Fresno State - PowerPoint PPT Presentation

Lance Burger Math 149 Fresno State. Teaching Mathematics from Necessity: Developing Meaningful Proportional Reasoning from Familiar Knowledge. It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry.

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Math 149

Fresno State

### Teaching Mathematics from Necessity: Developing Meaningful Proportional Reasoning from Familiar Knowledge

• It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry.

• -Albert Einstein (1879-1955)

• By the way: What equation is Einstein famous for?

Who does not know that a horse falling from a height of three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...Galileo-Two New Sciences

Q. Why do large dogs seem to pant more than small dogs? three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

FRONT & BACK three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

SIDES

TOP & BOTTOM

Reasoning from Manipulatives: three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

• Mathematical Concepts Related to LARGE and small dogs.

• Mathematical Concepts Related to HEAT EXCHANGE.

• How Can These Concepts be Compared Mathematically?

A demonstration about cooling: three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

Equal volumes of 100C water are placed in containers having different areas of exposure:

After several minutes measure the respective water temperatures with thermometers, significant differences in temperatures are seen.

A second cognitive disequilibrium of lesson: three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

• If more surface area produces better cooling, then why do large dogs appear hotter,…, they have more surface area than a small dog!

Reasoning from Manipulatives continued: three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

How can we compare surface area and volume AT THE SAME TIME?

As for the 200foot woman: three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

• A 5 foot woman made six times larger twice would reach 180 feet.

• (1/6)^2 is 1/36

• She would have 1/36 less skin as compared with the 5 foot woman.

• No wonder she is in such a small swimsuit … she is burning up!

p=8 three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

a=4

P=16

A=16

p/a=2/1

P/A=1/1

c=2(pi)r C=2(pi)2r three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

a=(pi)r^2

A=(pi)(2r)^2

=4(pi)r^2

c/a =2(pi)r/(pi)r^2=2/r

C/A=2(pi)2r/4(pi)r^2=1/r

three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...Knowledge develops as the solution to a problem’from Piaget’s Genetic Epistemology

Harel ‘s three or four cubits will break his bones, while a dog falling from the same height or a cat from a height of eight or ten cubits will suffer no injury?  ...

Necessity Principle:

For students to learn, they must see an (intellectual, as opposed to social oreconomic) need for what they are intended to be taught.

First figure out why you want the students to learn the subject and what you want them to know, and the method will result more or less by common sense (Feynman 1963).

Creating Lessons using Constructivist Needs-Based Design: subject and what you want them to know, and the method will result more or less by common sense (Feynman 1963).

• Begin with an Interesting Problem-Cognitive Disequilibrium.

• Edit out NON-NECESSITY

• STUDENT-HEAVY not TEACHER-HEAVY.

• Teacher as VOICE OF LOGIC.

• Classroom as LEARNING COMMUNITY.

• From PARTICLARS to ABSTRACT: RESOLVE disequilibrium with mathematical justifications.

CONSTRUCTIVISM subject and what you want them to know, and the method will result more or less by common sense (Feynman 1963).for mathematics

“students actively construct their mathematical ways of knowing as they strive to be effective by restoring coherence to the worlds of their personal experience.”

Cobb (1994)

It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry.

-Albert Einstein