[PPT] - Teaching modes of reasoning: Redesigning the Art of Approximation in PowerPoint Presentation

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Teaching modes of reasoning: Redesigning the Art of Approximation in Science and Engineering

Sanjoy Mahajan

MIT & Olin College

mit.edu/sanjoy/www/ sanjoy@mit.edu

HHMI Education Group, MIT, 27 February 2014

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Modes of reasoning or topics?

Doubling the block’s thickness changes the note frequency by what factor?

tap tap

a. 2

b. √ 2 c. 1

d. 1/

√ 2

e. 1/2

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Modes of reasoning or topics?

Doubling the block’s thickness changes the note frequency by what factor?

tap tap

a. 2

b. √ 2 c. 1

d. 1/

√ 2

e. 1/2

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A spring model explains the doubling in frequency

Compare the stored energies for the same deflection y:

y y

stored energy ∼ stiffness

k

×y2.

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A spring model explains the doubling in frequency

Compare the stored energies for the same deflection y:

y y

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A spring model explains the doubling in frequency

Compare the stored energies for the same deflection y:

y y

4× the energy per spring

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A spring model explains the doubling in frequency

Compare the stored energies for the same deflection y:

y y

4× the energy per spring 2× the number of springs

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A spring model explains the doubling in frequency

Compare the stored energies for the same deflection y:

y y

4× the energy per spring 2× the number of springs 8× the stored energy

stiffness×y2

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A spring model explains the doubling in frequency

Compare the stored energies for the same deflection y:

y y

4× the energy per spring 2× the number of springs 8× the stored energy

stiffness×y2

(bending) frequency ∼

stiffness

mass =

8×

2× = 2×.

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Modes of reasoning or topics?

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Modes of reasoning are a better organization than topics

Using modes of reasoning makes the course finite Using modes of reasoning promotes transfer Using modes of reasoning promotes long-lasting learning

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Topics are many, life is short

sound waves mechanical properties thermal properties weather fluid drag turbulence gravitation prime numbers retinal rod biomechanics astrophysics financial math . . . Where do you stop?

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Teaching in Cambridge, England, I moved toward modes

f reasoning only subconsciously

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I was influenced by problem solving in mathematics

The Invariance Principle Coloring Proofs The Extremal Principle The Box Principle Enumerative Combinatorics Number Theory Inequalities The Induction Principle Sequences Polynomials Functional Equations Geometry Games Further Strategies

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I used it for Street-Fighting Mathematics

Dimensions Easy cases Lumping Pictorial proofs Taking out the big part Analogy

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Modes of reasoning now seemed to appear everywhere

Pt. I. Incentives

Ex ante and ex post The idea of efficiency Thinking at the margin The single owner The least cost avoider Administrative cost Rents The Coase theorem . . .

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Modes of reasoning for science and engineering

rganized themselves slowly

Easy cases Divide and conquer Spring models Lumping Proportional reasoning Symmetry/conservation Abstraction Probabilistic reasoning Dimensional analysis

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Modes of reasoning for science and engineering

rganized themselves slowly

Abstraction Divide and conquer Easy cases Spring models Lumping Proportional reasoning Symmetry/conservation Probabilistic reasoning Dimensional analysis

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Modes of reasoning for science and engineering

rganized themselves slowly

Organizing Abstraction Divide and conquer Discarding Proportional Symmetry/conservation Dimensional analysis Probabilistic Easy cases Spring models Lumping

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Modes of reasoning for science and engineering

rganized themselves slowly

Organizing Abstraction Divide and conquer Discard: lossless Proportional Symmetry/conservation Dimensional analysis Discard: lossy Probabilistic Easy cases Spring models Lumping

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Modes of reasoning for science and engineering

rganized themselves around mastering complexity

lossless lossy proportional reasoning symmetry/ conservation dimensional analysis to master complexity lumping probability easy cases springs

rganize it

discard it abstraction divide/conquer

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Using modes of reasoning makes the course finite

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Modes of reasoning are a better organization than topics

Using modes of reasoning makes the course finite Using modes of reasoning promotes transfer Using modes of reasoning promotes long-lasting learning

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The tree gives each mode of reasoning a place

lossless lossy proportional reasoning symmetry/ conservation dimensional analysis to master complexity lumping probability easy cases springs

rganize it

discard it abstraction divide/conquer

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Each mode of reasoning contains examples: Divide and conquer

How many barrels of oil does the United States import in a year?

car usage Ncars 3×108 miles/year 20 000 miles/gallon 30 gallons/barrel 60 height 1 m width 0.5 m depth 0.5 m gallons/meter3 250

−1 −1

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Each mode of reasoning contains examples: Divide and conquer

How much energy does a 9-volt battery contain? battery life laptop power laptop-battery energy volume adjustment energy in a 9V battery

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Each mode of reasoning contains examples: Divide and conquer

How much energy does a 9-volt battery contain? battery life 4 hr (104 s) laptop power 20 W laptop-battery energy 2×105 J volume adjustment 1/10 energy in a 9V battery 2×104 J

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Using modes of reasoning promotes transfer

mode example

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Using modes of reasoning promotes transfer

When teaching by topics, it is too easy to use too-similar examples.

M p e n u m b r a

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Using modes of reasoning promotes transfer

Diverse examples help clarify the core idea.

M

Using modes of reasoning automatically produces diverse examples.

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Modes of reasoning are a better organization than topics

Using modes of reasoning makes the course finite Using modes of reasoning promotes transfer Using modes of reasoning promotes long-lasting learning

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The tree gives each mode of reasoning a place

lossless lossy proportional reasoning symmetry/ conservation dimensional analysis to master complexity lumping probability easy cases springs

rganize it

discard it abstraction divide/conquer

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Each mode of reasoning contains examples: Symmetry and conservation

force energy distance volume density v2 mass

−1

How fast will the cone fall? (by conservation of energy)

A volume Ad distance d v

drag force = energy consumed by drag distance traveled .

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Each mode of reasoning contains examples: Symmetry and conservation

force ρAv2 energy ρAv2d distance d volume Ad density ρ v2 mass ρAd

−1

How fast will the cone fall?

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Each mode of reasoning contains examples: Symmetry and conservation

How fast will the cone fall? F ∼ ρAv2 v ∼

F

ρA ∼

10−3 kg × 10 m/s2

1 kg/m3 × 0.01 m2 ∼ 1 m/s.

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The tree gives each mode of reasoning a place

lossless lossy proportional reasoning symmetry/ conservation dimensional analysis to master complexity lumping probability easy cases springs

rganize it

discard it abstraction divide/conquer

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Each mode of reasoning contains examples: Proportional reasoning

Wood blocks

tap tap

frequency ∝ thickness?

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Each mode of reasoning contains examples: Proportional reasoning

Falling cones again vfour stacked cones vone stacked cone =      4 2

r

√ 2 Equivalently, v ∝ (number of cones)?

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The tree gives each mode of reasoning a place

lossless lossy proportional reasoning symmetry/ conservation dimensional analysis to master complexity lumping probability easy cases springs

rganize it

discard it abstraction divide/conquer

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Each mode of reasoning contains examples: Dimensional analysis

Motto: The uncompared quantity is not worth knowing. cost of 9-volt battery energy cost of line (mains) power

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Each mode of reasoning contains examples: Dimensional analysis

Motto: The uncompared quantity is not worth knowing. cost of 9-volt battery energy cost of line (mains) power ∼ $1 / 2×104 J $0.15 / 3.6×106 J ≈ 7 × 180 ∼ 1000.

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Each mode of reasoning contains examples: Lumping

Every number is of the form:  

ne
r

few   × 10n, where few2 = 10.

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Each mode of reasoning contains examples: Lumping

How many seconds in a year? 365 days year × 24 hours day × 3600 seconds hour

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Each mode of reasoning contains examples: Lumping

How many seconds in a year? few×102 days year × 24 hours day × 3600 seconds hour

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Each mode of reasoning contains examples: Lumping

How many seconds in a year? few×102 days year × few×101 hours day × 3600 seconds hour

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Each mode of reasoning contains examples: Lumping

How many seconds in a year? few×102 days year × few×101 hours day × few×103 seconds hour

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Each mode of reasoning contains examples: Lumping

How many seconds in a year? few×102 days year × few×101 hours day × few×103 seconds hour ∼ few×107 seconds year

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The tree gives each mode of reasoning a place

lossless lossy proportional reasoning symmetry/ conservation dimensional analysis to master complexity lumping probability easy cases springs

rganize it

discard it abstraction divide/conquer

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Each mode of reasoning contains examples: Springs

Wood blocks

y y

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Each mode of reasoning contains examples: Springs

Why is the sky blue? How much energy does the earth–sun system lose in gravitational radiation?

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Connections are more important than facts alone

big cluster = 12% pbond = 0.40

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Connections are more important than facts alone

big cluster = 12% pbond = 0.40 big cluster = 42% pbond = 0.50

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Connections are more important than facts alone

big cluster = 12% pbond = 0.40 big cluster = 42% pbond = 0.50 big cluster = 81% pbond = 0.55

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Connections are more important than facts alone

big cluster = 12% pbond = 0.40 big cluster = 42% pbond = 0.50 big cluster = 81% pbond = 0.55 big cluster = 94% pbond = 0.60

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Using modes of reasoning promotes long-lasting learning

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Modes of reasoning are a better organization than topics

Using modes of reasoning makes the course finite Using modes of reasoning promotes transfer Using modes of reasoning promotes long-lasting learning

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Modes of reasoning are a better organization than topics

The goal [of teaching] should be, not to implant in the students’ mind every fact that the teacher knows now; but rather to implant a way of thinking that enables the student, in the future, to learn in one year what the teacher learned in two years. Only in that way can we continue to advance from one generation to the next. —Edwin T. Jaynes (1922–1998)

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