Download Quantitative Literacy (QL)

QR STEM Project
Dr. Robert Mayes
Science and Mathematics Teaching Center
QR STEM is a funded by a Department of Education Mathematics
and Science Grant (Project ID: 100150T2BA0).
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Three categories of Quantitative Reasoning
 Quantitative Literacy: use of number and
arithmetic to quantify a context with the goal
of understanding a phenomena so one can
make informed decisions.
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Quantitative Interpretation: ability to interpret a
model of a given phenomena with the goal of
understanding and making informed decisions;
algebraic, geometric, and statistical modes.
Quantitative Modeling: ability to create a model of a
phenomena with the goal of making predictions or
discovering trends.
Categories
Components
Quantitative
Literacy
Numeracy
• Number Sense
• Small/large Numbers
• Scientific Notation
Measurement
• Accuracy-precision
• Estimation
• Dimensional Analysis
• Units
Proportional Reasoning
• Fraction
• Ratio
• Percents
• Rates/Change
• Proportions
Basic Prob/Stats
• Empirical Prob.
• Counting
• Central Tendency
• Variation
Quantitative Interpretation
Interpreting
• tables
• graphs
• equations
• science models
• statistical plots
Logarithmic Scales
Statistics
• Normal Distribution
• Correlation
• Causality
Quantitative
Modeling
Logic
Problem Solving
Modeling
• linear
• polynomial
• power
• exponential
• conceptual models
• table or graph
Statistics
• Least Squares Fit
• Inference
• Hypothesis testing
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QL has four major components that
underpin the sciences:
Numeracy
 Measurement
 Proportional Reasoning
 Descriptive Statistics and Basic Probability
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Numeracy: ability to reason with numbers.
logic and problem solving aspect of QR on the arithmetic
level
 includes having number sense, mastery of arithmetic
processes (addition, subtraction, multiplication, division),
logic and reasoning with numbers, orders of magnitude,
weights and measures
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Number sense: awareness and understanding
about what numbers are, their relationships, their
magnitude, the relative effect of operating on
numbers, including the use of mental mathematics
and estimation (Fennel & Landis, 1994)

includes the concepts of magnitude, ranking, comparison,
measurement, rounding, degree of accuracy, percents,
and estimation
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Science Examples: diameter of a hydrogen nucleus is
approximately 0.000000000000001 meter while the total
energy consumption in the United States is
100,000,000,000,000,000,000 joules
http://learn.genetics.utah.edu/content/begin/cells/scale/
Scientific Notation: alternative representation
1×10 −15
 Hydrogen nucleus?
 U.S. Energy Consumption? 1× 10 20
Order of magnitude: How many orders of magnitude larger is U.S.
energy consumption then a hydrogen nucleus?
35
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Three techniques for bringing numbers into
perspective are estimation, comparisons, and
scaling
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Exemplar 1 (Bennett & Briggs, 2008): Provide
an order of magnitude estimate of how much
water U.S. citizens drink in a year.
Solution: Estimate that an average person drinks 3 10oz glasses of water
a day. There are 365 days in a year, and the U.S. population is on the
order 300 million or 3 x 108. So an estimate of water drunk is:
oz/year. There is approximately 0.3381 oz in a milliliter so an average person
drinks 9.716 x 1012 mL/year. The metric prefaces are another numeracy skill
that students must master for science, for example there are 1000 mL in a liter
so 9.716 x 109 L/year.
Exemplar 2 (Bennett & Briggs,
2008): Annual world energy
consumption is 5 x 1020 joules.
Many people are not familiar
with energy units like joules,
so make a comparison to
something that is familiar to
get a perspective on world
energy consumption.
Solution: One food Calorie is
equivalent to 4,184 joules. A
typical American uses 2,500
Calories of energy a day or
1.046 x 107 joules. So annual
world energy consumption is
approximately the same as
the daily energy use of 4.78 x
1013 people.
Exemplar 3 (Bennett &
Briggs, 2008): An
atom has a diameter
of about 10-10 meter.
Provide a scale that
puts this into
perspective.
Solution: If we multiply
the diameter by 1010
then we get 1 meter.
So there are 10 x 109 or
10 billion atoms in a
line along a meter
stick. A centimeter is
1/100 of a meter so
there are 108 atoms
along a centimeter
line, that is 100
million atoms.
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Science requires careful comprehensive measurement of quantities
such as distance, area, volume, discharge (1 acre-foot per day),
mass, density, force, pressure, work, moment, energy, power, and
heat.
Measuring is done with a variety of tools such as rulers, scales,
inclinometers, spectrometers, and fluorometers.
Measurement is sometimes direct and at other times is calculated
from other measures.
Fundamental characteristics of
measure are accuracy (how close
the measurement is to the actual
value), precision (how refined the
measure is), and error.
Exemplar 4 (Langkamp and Hull,
2007): The groundwater beneath
a gasoline station was
contaminated with methyl tertbutyl ether (MTBE), a gasoline
additive used to increase gas
mileage by increasing
combustion. MTBE is also a
cancer-causing agent. A
groundwater sample was
analyzed and MTBE measured
455 parts of MTBE per billion
parts of water. The threshold
measurement below which
MTBE cannot be detected is 1
part per billion. What can we
say about the accuracy and
precision of these measures?
Solution: Precision is high
since measure can
determine 1 part per
billion. Accuracy is
unknown since we do
not know the actual
amount of MTBE in the
water. Better to have
accuracy first, then
precision.
Exemplar 5
(Langkamp and
Hull, 2007): The
General Sherman
sequoia tree in
Sequoia National
Park has an actual
height of 83.82
meters. Using an
inclinometer and
trigonometry a
park ranger gets a
measure of 84.71.
What is the error
in the measure?
Solution: The absolute error
is 84.71 m – 83.82 m =
0.89 meters. The
relative error is 0.89
m/83.82 m = 0.0106 or
as a percentage 1.06%
error. Note there is no
unit attached to this
ratio since meters
cancel.
.
Exemplar 6 (Langkamp
and Hull, 2007): What
is the volume of water
an average household
uses each year to wash
dishes?
Solution: In this case an
estimate is about the best
for which we can hope.
Assume the household
does dishes once a day and
uses 6 to 10 gallons of
water per wash. Average
these values to get 8
gallons of water per wash.
There are 365 days in a
year, to make the
calculation easier round to
400 days. Then calculate
an estimated amount by
paying attention to the
units: .
.
Exemplar 7 (Langkamp and
Hull, 2007): A small well
in a rural village produces
3.5 gallons per minute.
What is this discharge in
m3/hour?
Solution: Dimensional
Analysis is the tracking of
units when performing
calculations. Requires
student to understand
ratios. Student can track
the calculations they need
to perform by tracking the
units. Students must also
have knowledge of units in
measurement systems and
conversions between units.
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Proportional Reasoning is a “form of mathematical reasoning that
involves a sense of co-variation and of multiple comparisons, and
the ability to mentally store and process several pieces of
information” (Lesh, Post, & Northern, 1988).
Pivotal position in science - most common form of structural
similarity, a critical aspect of recognizing similar patterns in two
different contexts.
Underpins many of the QL components, including measurement,
numeracy, and dimensional analysis.
The essential characteristic of proportional reasoning is to involve
reasoning about the holistic relationship between two rational
expressions (fractions, quotients, rates, and ratios).
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Proportional reasoning is not the ability to employ the cross
multiplication algorithm
Early phases in proportional reasoning involve additive reasoning
(A – B = C – D) and multiplicative reasoning (A x B = C X D).
Traditional proportional reasoning involves relationships of the
type A/B = C/D, where one of the values is unknown.
Karplus et al. (1983) views proportional reasoning as a linear
relationship between variables such as y = mx, where the yintercept is 0.
Proportional reasoning requires students to first understand
fraction a/b, which at the most basic level is interpreted by students
as comparing the part (numerator a) to the whole (denominator b)
for like quantities.
Solution: Percentage
difference is
calculated as
Exemplar 8 (Langkamp and Hull,
2007): Determine the
so we have
percentage of change in
bacteria in a lake with an
initial concentration of 720
colonies/liter and a final
concentration a week later of
1,260 colonies/liter.
Note that fractions do not have units, since the comparison is to the same object so the units cancel.
A basic QL skill is the ability to add, subtract, multiple, and divide fractions, as well as represent
fractions as decimals and percents.
Exemplar 9 (Langkamp and Hull, Solution: Ratio represents a
2007): The salinity of seawater
relationship between two
is typically expressed as
different quantities, focusing
number of grams or extra ions
on part-to-part comparisons.
in 1 liter of water. Seawater
Students must attain the
has about 35 grams of extra
conception of ratio before
ions per liter of water. What
being able to set up
is the salinity of common
equivalent ratios, one of the
seawater in parts per
fundamental conceptions of
thousand?
proportional reasoning.
Examples of ratios include
parts per thousand or
million, conversion factors
such as 1000 cm3/1 liter, and
normalizing data.
Exemplar 10 (Langkamp
and Hull, 2007): The
Little Snake prairie dog
colony in Colorado has
36,875 prairie dogs on
31,624 hectares, while
the Wolf Creek colony
has 20,009 prairie dogs
on 3,174 hectares.
Which colony is more
robust?
Solution: Normalize the data to
find the number per hectare
so you can make a
comparison on a common
scale. The density of prairie
dogs per hectare in the Wolf
Creek colony is
while the Little Snake colony has
only 1.17 pd/ha.
Exemplar 11 (Langkamp
and Hull, 2007): The
total area of tropical
forest in Congo is
278,797 km2 and in
Zaire it is 1,439,178 km2.
The protected tropical
forest in Congo is 12,935
and in Zaire 93,160.
What is the percentage
of protected forest in
Congo? What is the
percentage of total
forest in Congo to total
forest in Zaire?
Solution: The percentage of
protected forest in Congo is an
example of percentage as a
fraction, since we are comparing
part-to-whole for like quantities:
The percentage of total forest is a
part-to-part comparison, so it is
an example of percentage as a
ratio. Note we are still
comparing like quantities.
Exemplar 12 (Langkamp and
Hull, 2007): Carbon is
stored in various reservoirs
on Earth. The amount of
carbon in these reservoirs is
measured in petagrams
(Pg), where 1 petagram is
1015 grams. The amount of
carbon stored in fossil fuels
is 3,700 Pg while that stored
in vegetation is 2,300 Pg.
Using fossil fuels as a
referent, what is the
percentage difference
between carbon stored in
fossil fuels and carbon
stored in vegetation?
Solution:
So the carbon stored in
vegetation is 37.8% less than
that stored in fossil fuels.
Exemplar 13 (Langkamp
and Hull, 2007): In 1990
the forests of the world
covered 3,510 million
hectares. By 1995 world
forests had decreased to
3,454 million hectares.
How much forest will
be lost by the year 2010?
Solution: A pre-proportional reasoner
may use additive reasoning,
calculating change by taking the
difference in forest area without
accounting for the years over which it
occurs: 3,454 - 3,510 = -56 million
hectors. If they disregard the years
and consider this a yearly change
they would grossly overestimate the
amount of change. Calculating the
rate of change per year requires
finding the slope:
So over 20 years from 1990 to 2010 there will be a
loss of
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A proportion is an equivalence between two
ratios: a/b = c/d. Many students can manipulate
proportions to find the missing value, as in 2/5 =
x/10, however this may indicate only rote use of
the cross multiplying algorithm.
True proportional reasoning requires a perception
of structural similarity; a conception of n times as
many. If a student reduces A/B = C/D to P = C/D
when solving, then they are not using possible
structural relationships but are solving using
algebra without regard to structure.
Exemplar 14 (Langkamp
and Hull, 2007): A
capture-recapture
method is used to
determine the size of
the rat population on
an island. A sample of
250 rats is captured and
tagged. They are
released and allowed to
mix back into the
population. Sometime
later a random sample
of 500 rats is taken, and
21 are tagged. Estimate
the total rat population.
Solution: A student may simply set up a
proportion of tagged to total in the
sample and tagged to total in the
population
which using the cross multiplication
algorithm gives so the population is
about 5,952 rats. However such rote use
of the cross multiplication algorithm may
not indicate they are using proportional
reasoning. To understand why the
capture-recapture method works requires
the student to use proportional reasoning.
They understand that the population is
about 24 times as much as the tagged rats
and that this holds no matter how many
rats are tagged in the original capture.
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Student moves from a conception of
proportional reasoning as equivalent ratios to a
conception of linear direct variation y=kx.
Science contexts for proportional reasoning are
often in the linear direct variation form.
Students may extend this to indirect variation
y=k/x.
Requires an understanding of the underlying
algebraic concepts of equivalence, variable, and
transformations (structural similarity and
invariance).
Exemplar 15 (Rockswold, 2002):
Ozone in the stratosphere is
measured in Dobson units,
where 300 Dobson units is a
midrange value that
corresponds to an ozone
layer 3 millimeters thick. In
1991 the reported minimum
in the Antarctic ozone hole
was approximately 110
Dobson units. A 0 Dobson
measure would correspond
to 0 thickness of the ozone
layer. What is the direct
variation coefficient for a
model of the relationship
between Dobson units and
ozone layer thickness?
Solution: To show the relationship between
proportional reasoning and direct
variation, consider solving the problem by
setting up ratios of thickness to Dobson
units: 3mm/300Du and y mm/110 Du.
Now we have been told there is a direct
linear variation between the models, so
the ratio or slope is constant, so we can set
these ratios equal and solve for y, so y =
1.1 mm when the Dobson unit is 110. Take
any ratio of thickness to units, call it y/x.
Since the variation is linear we know that
this general ratio is equal to the constant
ratio of 3mm/300Du. Setting them equal
gives the proportion: y/x = 3/300 then
solving for y gives the direct variation
from of y = (1/100) x where 1/100 or 0.01
is the constant of variation.
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Probability is the chance of occurrence of an event,
with the theoretical probability defined as:
Earth systems cannot be manipulated like dice to
determine a theoretical probability. Often
scientists can only estimate the probability through
observations of the system. Empirical
(experimental) probability is determining a
probability based on observations or experiments.
Exemplar 16 (Langkamp and
Hull, 2007): From 1900 to
1998, there were 26 years in
which a major flood
occurred on the Mississippi
River. What is the
probability of a major flood
on the Mississippi River in
any given year?
Odds for an event are the ratio between the
event occurring and the event not occurring.
For Exemplar 16 the odds of having a major
flood in a given year are 0.26/(1-0.26) =
0.26/0.74 or about 1 in 3.
Solution: The event is a major
flood in a given year, which
occurred 26 times. The total
outcomes is the number of
years in which a major flood
could have occurred which is
99 (must count the year
1900). So the empirical
probability of a flood is
26/99 or approximately 26%.
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Descriptive statistics allow us to summarize
and describe data. The fundamental
descriptive statistics are measures of the center
of a distribution and measures of the spread in
a distribution.
Measures of central tendency include the
mean, median, and mode.
Variation is a measure of how much the data
are spread out. These include range, quartiles,
5 number summary, and standard deviation.
Year CRT Maker
Made
Lead
(mg/L)
Year
Made
CRT Maker
Lead
(mg/L)
Year
Made
CRT
Maker
Lead
(mg/L)
90
84
Clinton
Matsushita
1.0
1.0
93
84
Toshiba
Matsushita
3.2
3.5
94
77
Zenith
Zenith
21.5
21.9
85
Matsushita
1.0
84
Sharp
4.4
87
NEC
26.6
87
Matsushita
1.0
98
Samsung
6.1
96
Orion
33.1
89
86
84
94
94
97
Samsung
Phillips
Goldstar
Sharp
Zenith
Toshiba
1.0
1.0
1.5
1.5
1.6
2.2
95
98
89
97
87
98
Samsung
Chunghwa
Panasonic
Toshiba
NEC
Samsung
6.9
9.1
9.4
10.6
10.7
15.4
85
92
84
92
85
93
Sharp
Phillips
Quasar
Toshiba
Toshiba
Panasonic
35.2
41.5
43.5
54.1
54.5
57.2
97
91
KCH
Chunghwa
2.3
2.8
92
97
Chunghwa
Chunghwa
19.3
21.3
89
89
Samsung
Hitachi
60.8
85.6
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Solution: The median is the middle number when the data are
arranged in ascending order. If there are an even number of data
values then the median is the average of the two middle numbers.
There are 36 data values so we average of the 18th and 19th value:
The mean is the sum of all the values divided by the number of
values:
The mode is the most frequently occurring value, which is 1.0. The
mode is not often used in analyzing scientific data.
Notice that the mean and median values differ significantly, so it does
make a difference what measure of central tendency is reported. The
CRT makers may report the median to argue that the amount of lead
is not as high, while an environmental group would report the mean.
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The simplest measure of variation is the range which is the
difference between the largest and smallest values in the data set.
For the CRT problem in Exemplar 17 the range is 85.6 – 1.0 = 84.6.
What are concerns with using range as measure of variation?
While easy to calculate the range can be misleading, since one
outlier can make it appear the data set is more spread then it is.
To avoid this one can use quartiles (values that divide the data set
into quarters) and the 5 number summary – lowest value, lower
quartile, median, upper quartile, and highest value. For the CRT
problem the 5-number summary is 1, 1.9, 9.25, 29.85, 85.6. This
indicates that the lowest 25% only varies from 1 to 1.9 while the
upper quarter varies much more, from 29.85 to 85.6.
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Standard deviation measures the average distance of all data
values from their mean.
So we find the deviation which is the distance of a value from the mean
 Square the deviation so that positive and negative deviations don’t cancel out
when adding them which could conceal spread
 Take the average of all deviations by summing them and dividing by one less
than the total number of data values (this is an adjustment for working with a
sample rather than a population). The result is called variance.
 But variance is in squared units and our original data is not squared, so we
square root the variance to get standard deviation, which is in the same units as
the original data. So the standard deviation for Exemplar 17 is:
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Chebychev’s Rule: for any set of data at least 75% of the data lie
within 2 standard deviations of the mean and at least 89% of the
data lie within 3 standard deviations of the mean.
Any data value that lies 3 or more standard deviations from the
mean is called an outlier and it is common practice to discard
them from the data set.
Hitachi (1989) model with lead at a level of 85.6 mg/L is an outlier
since the mean is about 19 and 3 times the standard deviation is
66: 19+66 = 85 < 85.6
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We have restricted QL to the realm of number
and arithmetic, but variation requires algebraic
operations of taking roots or powers. So
variation is at best on the border of QL and QI.
We have discussed it in the QL section because
it is so commonly used to describe data.
Other basic statistics that are used in science
which are on this border between QL and QI
are z-scores (number of standard deviations a
data point lies above or below the mean) and
confidence intervals.