Transcript Plan for today - Trinity College

Plan for today
• Challenges of education/assessment
for QL
Challenges of QL Education &
Assessment
• QL is difficult – words & numbers, contexts,
unpredictability, ill-defined problems
• QL cannot be taught – habit of mind must be
learned, must be practiced, productive
disposition
• QL seems to require immersion across the
curriculum
• Canonical problem situation – read, interpret,
model, solve/compute/represent, reflect, &
critique
• Students have anti-QL habits formed in
traditional mathematics classes
The New York Times - October 14, 2001
One big advantage…
we are surrounded by sample problems…
we just have to learn how to educate for
solving them and to assess the resulting
learning.
Another advantage ..
we worry about how to educate for QL…
so we should rely on assessment of
learning to guide our work.
QL Assessment Questions to Consider
• What are the learning goals for QL?
• How can students' progress towards QL objectives
be assessed?
• What are the developmental steps in QL?
• What can current standardized tests tell us about
students' quantitative literacy?
• What should we value, i.e. what should we score?
• What are the standards for proficiency?
• Can we assess whether or not students are inclined
to practice?
• How are mathematical and numeracy skills related?
• What knowledge is needed?
Grant Wiggins on assessing QL
• Requires assessment of complex
realistic, meaningful, and creative
performances – authentic tasks
• Authentic tasks require
– Construction of knowledge
– Disciplined inquiry
– Value beyond school
• Threatens all mainstream testing and
grading in all disciplines, especially
mathematics
QL is not a new problem, just a recasting
In 15th & 16th century England, according to Pat Cohen in A
Calculating People,
“Strange as it seems, commercial life both triggered
and then limited the development of numeracy in
England. The adoption of Arabic numerals and
arithmetic came as a result of the expansion of
commerce in the 16th century, but textbook writers
then decided that their subject was too difficult for
bourgeois lads to learn with any degree of
understanding. They tried to strip arithmetic to its
essentials, but in fact they cut it into incoherent bits
and made it an arcane subject, almost impossible to
learn.”
Canonical problem situation
• Glean out the relevant information.
• Have confidence to take up the challenge.
• Estimate to see if assertions are reasonable.
• Do the mathematics.
• Generalize the situation.
• Reflect on the results.
Mathematical Proficiencies Needed
• Calculate and estimate -- decimals and fractions
• Recognize and articulate mathematics
• Generalize and abstract specific mathematical situations
• Functions as process -- especially linear and exponential
growth
• Some facility with algebra
• Use calculators to explore and compute
• Other examples will require knowing about shape -geometry and measurement -- and about data analysis and
probability
Mathematical Proficiency
• Conceptual understanding of mathematical
concepts, operations, and relations
• Procedural fluency
• Strategic competence
• Adaptive reasoning
• Productive disposition, that is, the habitual
inclination to see mathematics as sensible,
useful, and worthwhile, coupled with a belief in
diligence and one’s own efficacy
From NRC Study Report, Adding It Up, Kilpatrick, Swafford, & Findell, 2001
Plan for today
• Changes needed
Issues with Traditional Courses
• Emphases on components not processes
• Lack of mental constructs in lower level
courses
• Lack of venues for continued practice beyond
the course
• Not organized like the real world
• Tend to degenerate to methods and
procedures
• Develop template exercises expectations
• Not enough ambiguity
• Not enough interpretation and reflection
Changes in Pedagogy
• Mathematics should be encountered in many contexts such
as political, economic, entertainment, health, historical, and
scientific. Teachers will require broader knowledge of many
of the contextual areas.
• Pedagogy is changed from presenting abstract (finished)
mathematics and then applying the mathematics to
developing or calling up the mathematics after looking at
contextual problems first.
• Material is encountered as it is in the real world,
unpredictably. Productive disposition is critical for the
students.
• Much of the material should be fresh -- recent and
relevant.
Changes in Pedagogy
• Considerably less mathematics content is covered
thoroughly.
• The mathematics used and learned is often elementary but
the contexts are sophisticated.
• Technology – at least graphing calculators – is used to
explore, compute, and visualize.
• QL topics must be encountered across the curriculum in a
coordinated fashion. If I can coach writing then
literature faculty can coach QL.
• An interactive classroom is important. Students must
engage the material and practice retrieval in multiple
contexts.
Plan for today
• Can one play the game?
Playing the game vs Individual skills
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Dribbling
Passing
Shooting free throws
Jump shot
Know rules
Know positions
Know screens
Know defense
But can you play the game?
Game of QL
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Compute
Estimate
Graph
Interpret graphs
Solve equations
Geometry - area & volume
Rates of change
Odds & probability
Averages
Etc.
• But, can you read the
newspaper?
Processes
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Reading
Problem solving
Modeling
Reasoning
Communicating
Critiquing
Reflecting
Comparing
Plan for today
• Assessment items
The sales price of a car is $12,590, which is 20% off the original price.
What is the original price?
$14,310.40
$14,990.90
$15,290.70
$15,737.50
$16,935.80
Solve the following equation for A : 2A/3 = 8 + 4A
-2.4
2.4
1.3
-1.3
0
If r = 5 z then 15 z = 3 y, then r =
Y
2y
5y
10 y
15 y
If y = 3, then y3(y3-y)=
300
459
648
999
1099
Mary is reviewing her algebra quiz. She has determined that one of her
solutions is incorrect. Which one is it?
2x + 5 (x-1) = 9
x=2
p - 3(p-5) = 10
p = 2.54
y + 3 y = 28
y=45
w + 6 w – 3w = 64
w=8
t – 2t – 3t = 32
t=8
Which of the following is not a rational number?
-4
1/5
0.8333333……
0.45
Square root of 2
If Sam can do a job in 4 days that Lisa can do in 6 days and Tom can do
in 2 days, how long would the job take if Sam, Lisa, and Tom worked
together to complete it?
0.8 days
1.09 days
1.23 days
1.65 days
1.97 days
Add 0.98 + 45.102 + 32.3333 + 31 + 0.00009
368.573
210.536299
99.9975
80.8769543
109.41539
41% equals:
4.1
.41
.041
.0041
.00415
At a certain high school, the respective weights for the following subjects
are Mathematics 3, English 3, History 2, Science 2 and Art 1. What is a
student’s average whose marks were the following: Geometry 89, American
Literature 92, American History 94, Biology 81, and Sculpture 85?
85.7
87.8
88.9
89.4
90.2
What is the median of the following list of numbers? 4, 5, 7, 9, 10, 12
6
7.5
7.8
8
9
A building measures 20 ft wide by 32 ft long. If it has a flat roof,
which must be covered by plywood sheets measuring 4 ft by 8 ft,
how many plywood sheets are needed to cover the roof?
A. 32
B. 20
C. 8
D. 16
If you drove your automobile 396 miles and used 18 gallons of
gasoline, what is your gas mileage in miles per gallon?
A. 22
B. 20
C. 16
D. 14
In 2000, the National Center for Educational Statistics reported the
number of public high school graduates by state for the 1995-1996
to 1999-2000 school years.
The total number of high school graduates in the United States for the
school year beginning in 1998 was roughly 2.5 million. Estimate the
percentage of students who were educated in Maryland and Virginia
combined.
A. 45.0%
B. 4.5%
C. 30.0%
D. 11.4%
E. 25.0%
Number of Public High School Graduates
80,000
70,000
60,000
50,000
Maryland
40,000
30,000
20,000
Virginia
DC
10,000
0
1995
1996
1997
Year
1998
1999
A McDonald’s customer ordered a burger for 79¢, fries for $1.19,
and a coke for $1.29. If the locality has a 5% sales tax,
approximately what is the total bill?
A. $3.00
B. $5.25
C. $4.00
D. $3.45
Percent
Change
of
Value
In
Stock
40
30
20
10
-10
-20
’99
1990 ’91
’92
’93
94 ’ 95
96 ’
‘00
‘01
|
97 ‘ 98
From the information provided about company XYZ, what appears to have
occurred?
A. The value of the stock reached its highest value in 1995.
B. The value of the stock was still increasing in 1995, but began to
decline the next year.
C. The value of the stock increased in 1995 and the value also increased
but at a slower rate in the next year.
D. There is not enough information provided to answer the question.
6)
While on vacation in Las Vegas Jim wants to send a postcard to his
grandmother in Fayetteville, AR. He remembers that her zip code
begins with either “72” or “73”. How many 5-digit zip code
possibilities begin with either “72” or “73”?
A. 2000
B. 1998
C. 1000
D. 999
E. 10,000
7) If your score on a test is decreased by 30% and then increased by
35%, is the final result more or less than the original score?
Explain your answer.
If you flip a coin five times, which of the following results is (are)
more likely? Least likely?
The possible results of the flips, H or T, are in order from first
flip to fifth flip.
Explain your answer.
HHHHT, TTTTT, HHHTT, THHTT, HHHHH
If you flip a coin five times, which of the following results is (are)
most likely? Least likely? Explain your answer.
all H
all T
3T and 2H
4T and 1H
4H and 1T
Sample tasks:
1. Can both of these views be correct? Explain.
2. In each graph there is a “bar” over $20,000 to $30,000. Do these two
bars represent the same quantity? Explain.
Changing Health Care Costs
Why are these graphs titled as “The Rise in Spending?”
Assume that US citizens spent $10 billion on prescription
drugs in 1989.
a) Use the information in the graph on page 25 of the NYT
article (8/11/02) to produce a graph of prescription
drugs spending for 1989-2000.
b) On the same axes produce a graph of the same $10
billion over the same period when subjected to
inflation.
c) Write 2-3 sentences interpreting the meaning of the two
graphs.
Refer to the December 6 letter to
the editor, Math skills aren’t
great.
a)
Find the increase in percent
proficient.
b)
Find the percent increase in the
percent proficient.
c)
Is the letter writer correct
that the original article was
wrong? Why
d)
Is the letter writer correct or
incorrect when he states, “going
from 1 percent proficient tyo 3
percent proficient is an increase
of 200 percent?” Why?
Refer to the December 6 letter to the editor, Math
skills aren’t great.
a)
Find the increase in percent proficient.
1% X=3%
x=300%
b) Find the percent increase in the percent proficient.
.01 x 3.00 or .01 x 300% = .03
c) Is the letter writer correct that the original
article was wrong? Why?
No, he did not correctly calculate the percent change.
d) Is the letter writer correct or incorrect when he
states, “going from 1 percent proficient to 3
percent proficient is an increase of 200 percent?”
Why?
No, it is an increase of 300%, not 200%
a)
1%
Find the increase in percent proficient.
3%
The percent proficient increased by two percentage points.
b) Find the percent increase in the percent proficient.
3%  1% 2
 2 2 100=200 a 200% increase
1%
1
c) Is the letter writer correct that the original article was wrong? Why?
The letter writer was correct, but he needs to calm down a bit. It was a
small, common mistake, but a mistake nonetheless.
d) Is the letter writer correct or incorrect when he states, “going from 1
percent proficient to 3 percent proficient is an increase of 200
percent?” Why?
He is correct. The editorial assumed that if the # tripled, it would mean it
increased by 300%. What the editorial forgot to do was add on to the
original # to the problem.
1%
2%
X2
+1% 3% It is the same reason why a number that doubles increases
only 100%
a)
Find the increase in percent proficient.
Increase in percent: 3% - 1% = 2% increase
b) Find the percent increase in the percent proficient.
Percent increase:
 3 1 

 100  200%
 1 
c) Is the letter writer correct that the original article was wrong? Why?
Yes, because if the percent increase was to be 300% like the original article
stated, the ending proficiency would need to be 4% instead of 3%.
Ex:
 4 1 

 100  300%
 1 
d) Is the letter writer correct or incorrect when he states, “going from 1
percent proficient to 3 percent proficient is an increase of 200 percent?”
Why?
The letter writer is incorrect in making that statement due to a misuse of
wording. The letter writer made an error in saying “increase of 200%,”
when he should have said “it’s a percent increase of 200%.”
This letter to the editor appeared in the Arkansas Democrat-Gazette
on April 9, 2002.
My children asked me how many ancestors and how
many acts of these ancestors they are responsible for after
reading and listening to the Razorbacks’ coaching dilemma.
They have been taught that they are responsible for
their own actions and sometimes the actions of their friends
or even their parents. They just want to know how far this
goes back.
My daughter had visited the slave ship exhibit at one
of our downtown museums and recognized a family name as
being a builder of slave ships back in the 1500s in Britain.
She also knew that another relative brought six slaves over to
Jamestown in the 1600s. How much was she going to have to
pay in retribution? Was she the only one responsible or were
there others?
Before this got even more out of hand, we decided to do the math.
Assuming four generations per century and only one child per
family, that would be 19 generations. Two to the power of 19
would be 524,288 people who shared the responsibility.
Then we started laughing at the total absurdity of the idea
of one person today paying for the sins of another when there had
been 524,288 people in between.
And that wasn’t even counting brothers and sisters.
Conclusion: Get a life. Forgive and forget all 524,288 of
them.
Analysis of the argument
1. How might the writer arrive at the conclusion of 524,288?
2. Are there other possible conclusions for the number of
people in between the ancestor and the daughter?
The very forces that now press colleges to
address issues of quantitative literacy
were created by colleges and universities
in the first place. Changes in society that
demand widespread quantitative literacy
arose in large measure from innovations of
college graduates seeking greater
effectiveness and efficiencies in the fabric
of life and work. We must turn our
creativity toward coping with our
creations.