teaching quantitative literacy northeast consortium on quantitative literacy 2007 william briggs...
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Teaching Quantitative Literacy
Northeast Consortium on Quantitative Literacy 2007
William BriggsUniversity of Colorado
at Denver
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“...practically no one knows what they’re talking about when it comes to numbers in the newspapers. And that’s because we’re always quoting other people who don’t know what they’re talking about, like politicians and stock-market analysts.”
Molly Ivins, syndicated (U.S) columnist
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Mathematics
Statistics
BusinessManagement
Social Sciences
MedicinePublic Health
PhysicsChemistry
Computer ScienceArtificial Intelligence
BiologyEcology
Arts and Music
EconomicsPolitical Science
Engineering
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Mathematics for ...
CollegeCore requirementsNatural and social sciences coursesPre-service teachers.
Careers
Broad cross-disciplinary critical thinkingReal world problem solvingExplanation and presentation
Life
Personal : finance, taxes, gamblingCitizenship : economy, health, environment, voting
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Content Areas
Critical thinking, logic, problem solving
Informal fallacies Formal fallacies Argument analysis Organized, multi-step thinking
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Content Areas
Number sense and estimation Number perspective
Numbers in the news
Units
Uses and abuses of percentages
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Content Areas
Statistical reasoning and probability Surveys and statistical studies Interpreting graphs Causality Data analysis Lotteries, drug tests, health studies Assessing risk
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Content Areas
Linear and exponential growth The price of movies increases at a rate of
$0.75 per year. The cost of living increases at a rate of 3%
per year.
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Content Areas
Everything else !! Financial matters Mathematics and the arts Graph (network) theory Voting systems Apportionment Federal budget Energy
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(A. Tversky and D. Kahneman)
Linda is 31 years old, single, outspoken, and very bright. As a student, she majored in philosophy and was deeply concerned with issues of discrimination and social justice. She also participated in antinuclear demonstrations. Which is most likely?
• Linda is a bank teller.
• Linda is a bank teller and is active in the feminist movement.
Critical Thinking
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Ballot Initiative: Amendment to the Colorado Constitution (1992)
Shall there be an amendment to the Colorado constitution to prohibit the state of Colorado and any of its political subdivisions from adopting or enforcing any law or policy which provides that homosexual, lesbian, or bisexual orientation, conduct, or relationships constitutes or entitles a person to claim any minority or protected status, quota preferences, or discrimination?
Critical Thinking
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Numbers in PerspectiveU.S. Vital Statistics
3,960,000 births per year
13.7 births per 1000 people per year
7.5 births per minute
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Numbers in Perspective U.S. Vital Statistics
654,092 deaths per year from heart disease
29 deaths per 100,000 people
1.2 deaths per minute
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Percentages in the NewsTeen-age smoking rates are still lower than in the 1970’s. But the percentage of 12th grades who smoked daily last year jumped 20% since 1991, to 22 %, according to the most recent edition of the University of Michigan’s Monitoring the Future Survey,… The rate among 10th graders jumped 45%, to 18.3%, and the rate for 8th graders is up 44%, to 10.4%.
New York Times (4/20/97)
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Percentages in the News
The following correction appeared in the July 6, 1996 New York Times:
“A headline in some editions July 4 incorrectly characterized Boris Yeltsin’s margin over Gennady Zyuganov in the Russian presidential runoff. Yeltsin won by 13 percentage points, but had 33 percent more votes than Zyuganov.”
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Bloopers!
“It appeared that no fewer than 1 out of 25 cannot read or write ... 1 out of 25, which means 25% of our citizens.”
Dutch newspaper
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Bloopers!
“Tornado deaths from 1987 to 1996 dropped 300 percent from the 1940s when 1,176 deaths were attributed to tornadoes, an average of 179 per year.”
March 23, 1998 Associated Press
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Bloopers!
“By installing a metal halide fixture, you will get a $50 rebate and savings of “200% on energy.”
Northeast Utilities Advertisement (from A.K. Dewdney, 200% of Nothing)
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AA CEO
March 2003. American Airlines CEO takes 85% salary cut to avert budget crisis. The cut reduces his salary to $500,000. What was his original salary?
a. 1.85 × $500,000 b. $500,000/0.85
c. 1.15 × $500,000 d. $500,000/0.15
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AA CEO
Write a sentence:
Old salary – 85% of old salary = $500,000
Not:
Old salary = $500,000 + 85% of new salary
Simplify:
100% of old salary – 85% of old salary = $500,000
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AA CEO
Simplify:
15% of old salary = $500,000
Rewrite:
0.15 × old salary = $500,000
Solve:
Old salary = $500,000/0.15 = $3,333,333
Check!!!!
$3,333,333 – 85% of $3,333,333 =$500,000
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Bloopers!
“According to Lancaster Insurance, five out of four drivers between 17 and 21 have some sort of accident. The figure is correct because some have two accidents.”
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Bloopers!
“In the study, men who began taking light exercise in their sixties reduced their chances of dying by about 45% compared to those who stayed inactive.”
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Suppose that 1000 people are given a drug test that is 98% accurate and that 50 of the people actually are drug users. What percentage of the positive tests are false positives (nonusers who test positive)?
Drug Test Accuracy
• 50 are real users and could be falsely classified as non-users (false negatives)
• 950 are real non-users and could be falsely classified as users (false positives)
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Drug Test Accuracy
Accurate Error Total
Nonusers(true negative) (false positive)
Users(true positive) (false negative)
Total 1000
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Drug Test Accuracy
Accurate Error Total
Nonusers 931(true negative)
19(false positive)
950
Users 49(true positive)
1(false negative)
50
Total 980 20 1000
Percentage of false positives among all positives = 19/68 = 28%
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Mammogram Probabilities
Fact: Probability of a false report on one mammogram is 0.07.
Question: What is the probability of at least one false report on 10 mammograms?
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Mammogram Probabilities
p = Prob of false positive on one mammogram = 0.07
1 – p = Prob of no false positive on one mgm
= 0.93
(1 – p)10 = Prob of no false positive on 10 mgms
= 0.48
1 – (1 – p)10 = Prob of at least one false positive on 10 mgms
= 0.52
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Music and Exponential Growth
Frequency doubles every 12 steps
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Music and Exponential Growth
0
12
0
260 cps (Middle C)
520 cps (C above middle C)
for 0,1,2,...nn
F
F
F F a n
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Music and Exponential Growth
??? What is a ???
12
12
2
2 1.05946...
a
a
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Increase in College Tuition
0
2
4
6
8
10
12
14
1987 1988 1989 1990 1991 1992 1993 1994 1995
Per
cen
t In
crea
se CPI
Public
Private
Statistical Displays – College Tuition
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Statistical Displays
Death rates for various diseases: 1900-2000 (per 100,000)
0
100200
300
400500
600
700800
900
Pneumonia
Cardiovascular
Tuberculosis
Cancer
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Statistical Displays
Ratio of men to women
0.5
0.6
0.7
0.8
0.9
1
1.1
Years
Ra
tio All ages
<5
65+
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Population Pyramid
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Probability and the Law
An eyewitness whose reporting accuracy is 80% claimed that the taxi involved in a hit-and-run accident was Blue. There are only two taxi companies in town: 85% of taxis are Green and 15% are Blue. Based on this report, what is the probability that the taxi in the accident was Blue?
Consider 100 cases
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Probability and the LawCorrect (80%)
Wrong (20%)
Total
Blue 12 3 15
Green 68 17 85
Total 80 20 100
P(taxi was Blue given witness reported Blue) =
12 120.41
12 17 29
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Periodic Drug Dosing
You take 100 milligrams of antibiotic every 12 hours.
The half-life of the drug in your blood is 12 hours; that is, every 12 hours, the amount of drug in your blood decreases by 50%.
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Periodic Drug Dosing
0
1 0
2 1
1
100 mg
(0.5 ) 100 150 mg
(0.5 ) 100 175 mg
(0.5 ) 100
???????????????????????
n n
D
D D
D D
D D
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Periodic Drug Dosing
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9
D o s e n u m b e r
Am
ou
nt
of
dr
ug
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Finance – Savings Plans
Melanie opens a savings plan that offers an APR of 7.5% and she deposits $100 per month. Mort opens a savings plan that offers an APR of 9.5% and he deposits $80 per month.
After 20 years, which account has the larger balance?
How much has each person deposited?
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Finance – Savings Plans
Melanie has a balance of $55,373 and has deposited a total of $24,000.
Mort has a balance of $56,953 and has deposited a total of $19,200.
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ProjectsIs it safer to drive or fly?
Who benefits from a tax cut?
Are lotteries fair?
Can you detect a fraud? (Benford’s Law)
Should you believe a statistical study?
How do bar codes work?
How many words did Shakespeare know?
Is DNA fingerprinting reliable?
What is the Consumer Price Index?
Will cancer overtake heart disease (as the leading fatal disease)?
Is home-field advantage real?
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From The Wreck of Time, by Annie Dillard Harper's. vol. 296, no. 1772. Jan. 1998. pp. 51-56.
One tenth of the land on earth is tundra. At any time, it is raining on only 5 percent of the planet’s surface. Lightning strikes the planet about a hundred times every second. The insects outweigh us. Our chickens outnumber us four to one.
One fifth of us are Muslims. One fifth of us live in China. And every seventh person is a Chinese peasant. Almost one tenth of us live within range of an active volcano. More than 2 percent of us are mentally retarded. We humans drink tea – over a billion cups a day. Among us we speak 10,000 languages.
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From The Wreck of Time, by Annie Dillard
Every 110 hours a million more humans arrive on the planet than die into the planet. A hundred million of us are children who live on the streets. Over a hundred million of us live in countries where we have no citizenship. Twenty-three million of us are refugees. Sixteen million of us live in Cairo. Nearly a thousand of us a day commit suicide.
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From The Wreck of Time, by Annie Dillard
HEAD-SPINNING NUMBERS CAUSE MIND TO GO SLACK, the Hartford Courant says. But our minds must not go slack. How can we think straight if our minds go slack? We agree that we want to think straight.