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[1] Crisis --> Opportunity
"Within a crisis there are opportunities," is a cliche that clearly rings true
with respect to college algebra. Traditional college algebra is the crisis for
several reasons including a. A morally unacceptable low pass rate. b. Failure to meet its objective of inspiring and preparing students to enter a calculus tract. c. Failure to serve as a general education course.
d. Generates a culture of negativity with respect to mathematics.
This crisis is enhanced by the fact that college algebra is a college
gateway course--a gateway that the traditional course closes, as a
result of its high failure rate, to over half a million students per
year. The opportunity is to refocus the traditional course to form a bridge linking what Lynn Steen and Bernie Madison call the two mathematics, formal mathematics and quantitative literacy (QL). The refocused course is a modeling course that emphasizes problem solving in the modeling sense rather than in the traditional exercise sense. This means that problems are set in a real world context and solutions are interpreted in light of the real world setting as illustrated in the following diagram.
Problems are presented in story form for as Scott Hunt, a retired engineer
from Scott Paper Company, said "In the real world, there are only story
problems." Formal techniques are introduced when needed. The philosophy is to
leverage the power of human reasoning to analyze and interpret while using the
power of technology to aid in computation.
The primary pedagogical goal of the refocused course is to develop students to
be exploratory learners who take responsibility for their own learning and
extend their efforts beyond the confines of the course. Developing an
inquisitive approach and a habit of mind that seeks quantitative evidence to
make an argument are essential ingredients of exploratory learning.
The emphasis on developing communication skills--reading, writing,
presenting--is another aspect that contrasts a refocused college algebra
course from a traditional course. Students learn how to transform a written
description (story problem) into mathematical form, solve the "mathematical"
problem, and then interpret the results in light of the original context. In
addition to studying the text (reading) and preparing written reports,
students make frequent class presentations.
Small group work is also central to the refocused program. Students are
engaged in small group, in-class activities on a daily basis in addition to
working at least two out-of-class projects. [2] Age DependencyAge dependency ratios are used in marketing, developing social security legislation, housing projections, etc. Using the 2000 population data in the following table, compute the following dependency ratios for the United States and Haiti. \The age dependency ratio is derived by dividing the sum of the 0-to-19 and 65-and-over populations by the 20-to-65 population and multiplying by 100. \The old-age dependency ratio is derived by dividing the population of 65 and over by the 20-to-64 population and multiplying by 100.
\The child dependency ratio is derived by dividing the population under
0-to-19 by the 20-to-64 population and multiplying by 100. Estimates of the 2000 population distributions in the three age categories: 0-to-19, 20-to-64, 65 and older for the United States and Haiti. Source: U.S. Census Bureau www.census.gov.
Recognizing that the United States is the richest country in the world and
Haiti is one of the poorest countries, what conjectures can be drawn by
comparing their age dependency ratios? What implications for the future can
you conjecture based on your dependency
ratios? [3] Oil Heating Costs
The price of heating oil has increased almost five-fold since 2002 with the
largest increase occurring this year (58%). This recent increase is
threatening millions of people with an impending home heating crisis this
coming winter. The Bangor (Maine) Daily News, August 8, 2008, published the
following table
Display the year/price data in a bar chart. Fill in the third column and then
display the Winter/Annual Increase in a bar
chart. [4] Queriesa. Pick a number, add 3 to it, double the result, subtract 4 from the answer, and then triple what you get. If the answer is 39, what was the original number? b. How far out of your way would you drive to save six cents per gallon on the price of gasoline? Clearly state your assumptions and explain your reasoning. c. If inflation over a given year was 300 percent and if a shovel cost $25 at the start of the year, how much would it cost at the end of the year?
d. If prices double in one year, is the inflation rate 100 percent or 200
percent? Explain your
reasoning. [5] Take a Good Look, using Math
"Wells' career short, but robust" Jacqueline Brannon Giles
Texas Southern University
Forty-six years ago a young man greeted me in the hallway of the Nabrit
Science Building at Texas Southern University, Houston, Texas. He was on his
way to Mrs. Corinne Newell's mathematics class and I was taking a break,
walking down the hallway. That moment began an adventure in collaboration and
I am taking a good look at the people and experiences of the past.
The man, an elementary education major, who spoke to me in the hallway at
Texas Southern is Warren Wells. He became a professional football player, a
wide receiver, while I became a mathematics professor. After re-encountering
him in February 2007, I began to search for some way to measure the impact of
his career. Unfortunately, I did not archive statistics on the National
Football League (NFL) or on his career. On May 18, 2008, however, a sports
enthusiast and researcher published a quantitative analysis of thirty-three of
the NFL wide receivers, starting with Don Hutson (January 31, 1913 -- June 24,
1997). Warren Wells, a veteran who is now 65 years old, was included. I
scanned the data and saw that Wells ranked No. 1 in two categories, and it
looks like he has held that rank for about 38 years
I decided to compare Warren Wells with Jerry Rice by analyzing the plot of the
data for the two wide receivers. Letting the left most, lower corner of the
chart be (0,0), I let the vertical axis represent ranking and the horizontal
axis categories (spaced at 5 point intervals). So, if Jerry Rice, an NFL wide
receiver, ranked 5 in the fifth subcategory, there was a point in his data set
of (5, 5). Data points were plotted and a line graph was drawn for both Jerry
Rice and Warren Wells. The next step was to turn the problem over to my
calculus class and ask the students to calculate the area under each of the
line graphs. One of my students who received his law degree from Harvard
University questioned the arbitrariness of the spacing on the horizontal axis.
The students finally concluded that the spacing could be arbitrary as long as
it was uniform and consistent for both players. The students became excited
because they were using calculus to analyze historic NFL data, and they were
totally surprised that the mathematics professor was talking about great wide
receivers. The students began to take a good look at the NFL data, using
mathematics.
What were the results of the study? Well, the students concluded that overall
Jerry Rice ranked higher than Warren Wells, although Warren Wells ranked
higher in two subcategories. His rank is No. 1 in "Yards/Reception" and in
"Yards/Attempt". He has held the top rank for 38 years, which is longer than
Jerry Rice's career of 20 years. Using calculus, the total area under Rice's
curve was found to be greater than the area under Wells' curve. The difference
in the areas was not that large. Also, although Jerry Rice's overall average
rank is greater than Warren Well's, Rice is ranked No. 1 in only one category,
"TDs/Game." In the three categories for "Rushing" Warren Wells, Don Hutson and
Jerry Rice all rank No. 1.
Hutson, the first NFL wide receiver, dominates "Yards/Game" while Rice
dominates "TDs/Game" and Wells dominates "Yards/Attempt". The alumnus from
Texas Southern University has career statistics that are shining bright in
2008. My students and I encourage you to take a good look at the data on www.behindthesteelcurtain.com/2008/5/18
/520032/quantitative-analysis-of-t and decide whose career statistics stands
out over a long time period. You may be surprised at your results. Here is a
short report based on student input:
Calculus students analyzed a quantitative analysis comparing the career
statistics of Jerry Rice and Warren Wells. They created a model for each
player by graphing a piecewise defined function connecting ranks and then used
integration to find the area under the piecewise linear curves. The students
asked, "Why hasn't Wells been considered as a nominee to the NFL Hall of Fame
since he ranks No. 1 in two of the seven categories in the quantitative
study?" They noted that the statistics indicate that Wells has ranked No. 1 in
two categories for about 38years. Source: WolfpackSteelersFan on May 18, 2008
7:42 PM EDT
The students in the project group were: Shannon Harrison, Jeff Sickorez,
Ashely Tish, and Teyshia Waters. Shannon Harrison, an electrical engineering
major and math minor, was the group
leader. [6] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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