Category Archives: SCIENCE AND MATH

Mathematics: Curricular Overview

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Concept Example #1 Example #2 Example #3
Geometry: power of deductive reasoning Thales and pyramid of Giza Eratosthenes and circumference of earth Chinese proof of Pythagorean theorem
Statistics: chance Birthday Problem Odds of rolling a 3 versus a 7 Simpson’s Paradox

Bayes Rule

Base Rate Problem
Exponential Growth The rice on the chess board Paper folding Inflation
Calculus: optimization The optimal size of  a can The camel problem,

Dido’s problem

 The pipeline problem
Magic Numbers Pi  – what is so special? The mystery of e

 

 Can you visualize i?
Scaling First power- lengths Second power–areas Third power

volumes
Counting Taxes + virtues +

Income +

 Cooking ingredients Good deeds per day; smiles; kind words
Calculating Ratios: santa/Grinch

Real versus reported earnings

 Real versus nominal returns, pre tax versus after tax returns Discounted versus non discounted cash flows or reported debt

And real debt
Ethics of mathematics Is economic inequality a measure of injustice? How to distinguish between factors behind ethnic disparities? How to measure factors behind gender disparities?

What is a fair interest rate to charge a sibling?

 

What is the biggest hole you can cut into an 8 by 11 piece of paper?

How many piano tuners in Chicago?

How does Social Security work?

How does the welfare system work? (EITC, minimum wage, TANF, Medicaid)

What is the difference between the Federal budget deficit, the Federal debt, and the real debt of the federal government?

What is the difference between real and nominal interest rates and return?

What is bracket creep? Why are benefit payments inflation-adjusted and tax brackets are not?

Which numbers are highlighted and which ignored by proponents of a flat tax?

Physics – Curricular Overview

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Key Concept Example A Example B Example C
Leverage See Saw, pulley Baseball       Debt
Gravity Aristotle and

Galileo’s Pisa Experiment; three ramps where does each ball end up?

 How gravity dictates how to stand, walk, sit,

 

 Galileo’s inclined plane and measuring gravity
Temperature

And Pressure

Differentials

 The Sea Breeze

Global atmostpheric circulation

 Phase Diagrams:

Water, carbon

(sweaty soda can)

Mountains make rain and deserts

 The human body: pleasure, pain, deep sea diving,

airplanes
Angles matter Flight The seasons

Eclipses

 Photography
Action and Reaction The astronaut cut off from space ship Flight – not just about Bernouilli The balloon
Inertia Why we need seat belts Newton’s Apple and the Moon Why seats in troop transport planes face backwards
Entropy The dorm room The broken egg      Car crash
Free Body Diagram bridge Airplane in flight Gravity defying straw
Fundamental Units length mass time
Shapes matter wheels Triangles Arches

 

Agenda for Physics Teachers:

 

Identify the best three museum exhibits from anywhere in the world for each concept worth remembering.

Identify most productive science teacher by subject and three best experiments she uses to demonstrate it.

Identify the minimum number of around the house items needed to illustrate each concept.

How few items can you carry around in a backpack to teach the lion’s share of the physics curriculum?

Identify the physics experiments that usually fail and stop doing them.

Identify the physics experiments that work and make sure each student can teach them.

How much does the earth weigh? How far is the sun? What is the circumference of the earth? Geometry and the power of deductive reasoning.

The miracle of the circle, the miracle of the triangle—the math of the universe.

Chemistry: case study in analysis, synthesis, precision of measurement, meticulous recording

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Example #1 Example #2 Example #3
Essential Nutrients

And why we

Need them

  

Amino Acids

  

Calcium

  

Iron
Essential drugs and how they work  

Penicillin

  

Excedrin

  

Zantac
Dangerous drugs

why people use them and why we should avoid them

  

Alcohol

  

Heroin

  

Cocaine
Dangers of Household Chemicals  

Ammonia and

Bleach

 

  

Sulfuric Acid

In car Batteries

  

Insecticides
 

Chemistry of the Earth

 

  

Oxygen —

 

 

  

Iron —

Key to magnetic field that protects life

  

Nitrogen
 

Assessing “scientific studies” of dangers or advantages of diet, drugs

 

  

Tobacco: when did the evidence cross significant threasholds?

  

Fat: why was fat considered bad and then that finding rejected?

  

Sugar and Gluten: what is the  evivdence? What does it mean?
 

How to control your own body’s

Production of chemicals

 

 Cortisol / Testosterone  

Dopamine

Endorphins

Oxytocin

Serotonin

 

  

Nitric Oxide

Leptin
 

Making Bread

 

  Yeast Baking Soda  Gluten

 

Physics: Case study Matrix – Flight

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The Angle of Attack Action/Reaction Pressure Differential
 

The Hand Outside the Car Window

 

  

The Balloon letting air out

  

Ping Pong ball and

Hair Dryer
 

Little Angles matter:

Tilt of earth, etc.

 

  

The Astronaut/Skater

  

Bernouilli’s Principle
 

The Ailerons

 

  

The Engines

  

Flying Upside Down

Shape of Wing

 

Quantitative Literacy

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                             Do the numbers really mean what they seem to?

  

     Race                         Gender                    Class

 

Unadjusted number that suggests discrimination or injustice.

 

  

Blacks 12% of pop 40% of prisoners

Or 50% of those stopped and frisked

  

Women make

$.77 on the dollar

  

Top 1% of households make 20% of income -suggests injustice
 

 

What other factors

might account for the differential?

 

 Could crime rate differentials account for the differentials?

Could the war on drugs itself not racism be the real problem?

Could family structure inequality be a factor?

  

Could preference for flexible hours,

or lower paying

care-giving professions

account for most of the difffential?

  

Should the numbers be adjusted for hours worked?

Workers per household?

Age?

Productivity?

So what?

 
 

Facts and questions

  

Women are 50% of pop only 5% of prisoners.

 

Sexism?

 Adjusting for these Harvard economist Claudia Goldin

finds the gap virtually non-existent.

 

Is she wrong?

 The top 1% pay 40% of income taxes – twice their share of income.

 

Is that fair?

Who decides?

How?

Is meritocracy bad?

Is the real problem equality of opportunity not inequality of income?

 

Statistical Literacy

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    #1                                #2                                #3

Descriptive
Statistics:Pictures are worth a thousand numbers as well as a thousand words.		 Why a histogram is better than a mean or median or even a five number summary of a set of data. Why a scatter plot is better than an Rsquared or a

Regression

Equation in summarizing the relationship between two sets of data.

 Judgment is key to adjusting the axes of the histograms and scatter plots to maximize the quality of information
The average American has one testicle and one ovary. Gathered data is not always good data Correlations are not causation Most important may be ignored by the analyst
Has the data been massaged? Are the outliers there? The most important facts may not be quantifiable. Problem sets should be prioritized by civic or personal relevance. Failure to do so is a recipe for amnesia, boredom, and poor performance.
Inferential Statistics:

 

All about randomness, probability, and sample size

 

 Randomness is key to getting a good sample The bigger the sample size the closer and more confident you can be in generalizing. Roughly: a random sample of 100: 95% confident, plus or minus 10%.

Sample of 1200” 95% confident

Plus or minus 3%
 

Beware the file drawer problem!

 

Beware Type 1 and Type 2 Errors

 Probability is the key to statistical experiments.

 

Has the experiment been reproduced?

How many times?

 Perfect analogy is to the jury system. As the jury should assume innocent, so the statistician assumes no effect

(null hypothesis)

 Then calculates odds of getting actual result from chance alone. If extremely rare then, rejects the null hypothesis
 

Data omission and factor omission are likely when issue has a partisan dimension.

 

  

P values are arbitrary.

P values should be stated a priori.

P values should be thought about.

  

Chi square calculations can be completely misleading.

 

  

Simpson’s Paradox is a warning to make sure all the data has been disclosed.

 
Finding Right Metric key Best hitter: is batting average the right number?

Is Z-score better than absolute?

 Finance: absolute or relative performance? risk-adjusted or not,

But how? Sharpe?

 Justice: do women make $.77 on the dollar? What does this mean? Are you sure?

 

Statistical Literacy -2

 

 

 

Level One

 The uncertain can often be predicted with amazing certainty. The laws of chance lead often to extremely counter-intuitive results. Data can be misleading and decisions based on them false.
Quantification can lead to the double illusion of importance and objectivity, The most important factors may not be quantifiable. Most complex problems require non-quantitive judgment.
Statistical wizardry is no substitute for substantive knowledge. Experiments should be reproduced multiple times. The bigger the sample the lower the standard deviation.
Level Two 1111 is a good sample size –

which is not a function of the population – the tasting soup analogy

 P values are arbitrary but should be decided on before experiments are conducted. For what is a p value of 5% a good decision rule? Guilt or innocence?
The inevitability of Type 1 and Type 2 errors Studies should be based on random samples.

 

 Experiments should be double blind and controlled.
Regression to the man, the Placebo effect, and the Hawthorne effect can be big Adjusting data is often necessary but can be extremely misleading. CPI adjustment is critical but fails to account for quality improvement.
Extrapolation is almost irresistible: budgets, stocks,

Climate.

 Partisan bias can distort data collection, experimental design. Only 40% of social science experiments are ever repeated.

Is this science?