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[1] Notes on Problem Solving, CUPM Curriculum Guide 2004
(The following is copied from the CUPM Curriculum Guide, 2004, p. 15. The
entire Guide is available on-line at www.maa.org/saum/.) Develop persistence and skill in exploration, conjecture, and generalization.
Problem solving requires more than just solid mathematical reasoning---there
are broad strategies and mental attitudes that students must identify, master,
and internalize. To be successful problem solvers, students must learn
persistence in the face of repeated rebuffs and flexibility in the choice of solution strategies. They must
replace the question "Did I get the correct answer? with the question "Does
my answer make sense?" Students must also learn to explore examples and
special cases, to let new knowledge lead to new questions, to generalize and
pose conjectures, yet to test all conjectures and retain a healthy skepticism
toward unproven claims.
Instructors can stimulate students to generate questions and comments in
response to readings, exercises, and presentations by modeling good
questioning behavior. What do the words mean? What are some non-trivial
examples? What motivates the material? What assumptions are being made? How do
I know this is right? Through careful choice of problems and dialogue with
students, faculty can also lead students to develop a more skeptical stance
toward assertions: Does this make sense? Have all assumptions been
enunciated?
Students need to be exposed to multi-stage projects that are built on
exploration and conjecture and require persistence and flexibility for their
solutions. More generally, at least some courses should be restructured to
shift the burden from instructor to students for discovering and justifying
results. A mathematical modeling course is an especially apt setting in which
to make this shift and to raise students' awareness of the need to state
problems carefully, articulate assumptions, and apply the mental and strategic
tools of problem solving. Indeed the value of modeling lies at least as much
in the artful and creative thinking and thoughtful interpretation that it
requires as in the connections it makes between mathematics and other
disciplines. [2] Book Review: Contemporary College Algebra: Data, Functions, Modeling
Written by Chris Arney, Vice President
The College of Saint Rose, Albany, NY
(This review appeared in Mathematics and Computer Education, vol 38,
no. 1, Winter 2004)
I consider teaching and learning college algebra big challenges for the
teacher and student, and last semester, I decided to try a reformed approach
to the subject. I used Don Small's textbook, which has as its philosophy to
educate students for the future. The textbook tries to help students become
exploratory learners by having them perform queries (just-in-time explorations
included within the text's chapters) and projects (data-driven applications
within the text's exercises). I chose this book because I liked Small's list
of goals, including improving communication skills, developing teamwork
skills, enhancing technology skills, empowering modeling, and building
confidence. I think the text helped my students in all these areas.
Don Small is a master at presenting fun and challenging problems. He has
included applied problems in sections called "Fun Projects," which require
student teams to research, model, solve, and write. Most of these problems
start with data that the students are required to turn into useful
information.
The book's content is divided into three chapters --- data and variables,
functions, and modeling. Small encourages the use of technology (mostly a
graphing calculator) to help students solve messy problems and the use of
graphical analysis in their problem-solving and analysis. These technology
skills were a challenge for my students and for me, but once we became used to
our calculators and their capabilities, we made real progress on these
skills.
In case you think this book is much different from others with the "College
Algebra" title, I will list some of the algebraic and arithmetic skills it
contains. Some of the classic techniques covered are: percentage, fractions,
radian measure, inequalities, graphing, transformations, function evaluations,
factoring, iteration, polynomials, logarithmic and exponential functions,
trigonometric functions, parametric functions, and logistic
functions.
Having used this textbook, I would say that college algebra continues to be
challenging. However, I believe that Small's text helped my students achieve
some of the book's goals and gave them a taste of exploratory learning,
inquiry, and modeling. I recommend trying this book. [3] Bald Eagle Project
The bald eagle, our national bird, is the only eagle native to North America.
The population of eagles, estimated to be as large as half a million when
European settlers began to arrive, steadily decreased reaching a low of
several hundred pairs in the 1930s. The effects of the Bald Eagle Act of 1940
to provide protection, were compromised by the rapid growth of the use of DDT
in agriculture. This chemical greatly reduced the birth rate as it weakened
the shells of the eagle eggs making them unable to withstand incubation. In
1967, the bald eagle was declared an endangered species and five years later
DDT was banned for use as a pesticide. This marked a turn around in the growth
of the eagle population as indicated by the following chart. (Source: U.S.
Fish and Wildlife Service.) In 1995, the bald eagle was elevated from the
"Endangered Species" category to the "Threatened" category.
The following questions can be used to develop a bald eagle project.
a. Form a scatter plot of the data and then using two points, fit a linear
curve to the scatter plot.
b. Use a calculator or computer to determine a linear regression
equation.
c. Compare the results from parts a and b. If they are different, explain what
caused the difference.
d. Interpret the meaning of the slope in the equation in either part a or part
b.
e. Use the regression equation to predict the number of eagle pairs in 2000
and in 2002.
f. Research the official population figures for bald eagles for 2000 and 2002.
Compare the results with your predictions in part e. Comment on any
differences between your predictions and the official figures. [4] Measuring for a Roof
(This real-life exercise is an example of the use of college algebra in the
workplace.)
Bob called the other night asking for help in determining the measurements for
a roof that he is building over his portable sawmill. The roof trusses are
triangular with a base of 27.5 feet and the two slanted sides have equal
lengths (isosceles triangle). The roof's elevation angle is 30 degrees. Bob
wanted to know the height and slant length of the trusses. Can you help
out?
An interesting follow on question is how many bundles of three-in-one asphalt
shingles will be needed to shingle the roof given that the roof is 13 feet
long and a bundle of shingles covers
[5] G-force and NASCAR Drivers
G-force is the gravitational force exerted on a body; one `G' is equal to the
Earth's gravity. G_force is modeled by the following function of three
variables:
a. At what time is the maximum velocity achieved by the driver? b. What is the maximum velocity achieved by the driver?
c. What is the maximum G_force experienced by the driver? [6] Test Questions
a. Create a scenario that can be modeled with a decreasing function that is
never zero and then sketch the function.
b. Create a scenario that can be modeled with a periodic function and then
sketch the function.
c. Dianna opened an ice cream shop in her house one block from the main street
of a small town. Sketch a graph showing her sales from April through August.
Explain your reasoning in drawing the graph in the shape that you did.
d. Determine a quadratic equation whose graph contains the points: (0,5),
(2,0), (4,3). [7] Dissemination Workshop
A three day Dissemination Workshop for the Contemporary College Algebra
program will be held May 26-28, 2004 at Miami-Dade College, Miami, FL. Norma
Agras (nagras@mdc.edu) is the organizer. Paul Dirks, Elizabeth Succo, Pavlov
Rameau, Alvio Dominques, Alex Fluellen, and Don Small will serve as
facilitators. Activities will feature: (a) Hands-on, small group activities and projects (b) Use of graphing calculators in teaching
and learning college algebra
(c) Problem solving in the modeling sense
(d) Modeling using recursive sequences
(e) Elementary data analysis
Several discussions are planned for the workshop. A sampling of topics include
the role of college algebra in a student's academic program; formulation of
goals for a college algebra course; the national movement to refocus college
algebra, and the use of technology. [8] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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