A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus gordonspfarmingdale.

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A Fresh Start for Collegiate Mathematics
Rethinking theCourses below Calculusgordonsp_at_f
armingdale.edufgordon_at_nyit.edu
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College Algebra and Precalculus
Each year, more than 1,000,000 students take
college algebra, precalculus, and related
courses.
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The Focus in these Courses
Most college algebra courses and certainly all
precalculus courses were originally intended and
designed to prepare students for calculus. Most
of them are still offered in that spirit. But
only a small percentage of the students have any
intention of going on to calculus!
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Enrollment Flows

• Based on several studies of enrollment flows into
  calculus
• Less than 5 of the students who start
  college algebra courses ever start Calculus I
• Virtually none of the students who pass college
  algebra courses ever start Calculus III
• Perhaps 30-40 of the students who pass
  precalculus courses ever start Calculus I
• Only about 10 of students in college algebra
  are in majors that require calculus.

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Why Students Take These Courses

• Required by other departments
• Satisfy general education requirements
• To prepare for calculus
• For the love of mathematics

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What the Majority of Students Need

• Conceptual Understanding, not rote manipulation
• Realistic applications and mathematical
  modeling that reflect the way mathematics is
  used in other disciplines
• Fitting functions to data
• Recursion and difference equations the
  mathematical language of spreadsheets

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• The Link to Calculus

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Calculus and Related Enrollments
In 2000, about 676,000 students took Calculus,
Differential Equations, Linear Algebra, and
Discrete Mathematics (This is up 6 from
1995) Over the same time period, however,
calculus enrollment in college has been steady,
at best.
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Calculus and Related Enrollments
In comparison, in 2000, 171,400 students took one
of the two AP Calculus exams either AB or BC.
(This is up 40 from 1995) In 2004, 225,000
students took AP Calculus exams In 2005, about
240,000 took AP Calculus exams Reportedly,
about twice as many students take calculus in
high school, but do not take an AP exam.
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Some Implications

• Today more students take calculus in high school
  than in college
• And, as ever more students take more mathematics,
  especially calculus, in high school, we should
  expect
• Fewer students taking these courses in college
• The overall quality of the students who take
  these courses in college will decrease.

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Another Conclusion
We should anticipate the day, in the not too
distant future, when college calculus, like
college algebra, becomes a semi-remedial
course. (Several elite colleges already have
stopped giving credit for Calculus I.)
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Who Are the Students?
Based on the enrollment figures, the students who
take college algebra and related courses are not
going to become mathematics majors. They are
not going to be majors in any of the mathematics
intensive disciplines.
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Associates Degrees in Mathematics

• In 2000,
• There were 564,933 associate degrees
• Of these, 675 were in mathematics
• This is one-tenth of one percent!

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Bachelors Degrees in Mathematics

• In 2000,
• There were 457,056 bachelors degrees
• Of these, 3,412 were in mathematics
• This is seven-tenths of one percent!

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Some Conclusions
Few, if any, math departments can exist based
solely on offerings for math and related majors.
Whether we like it or not, mathematics is a
service department at almost all
institutions. And college algebra and related
courses exist almost exclusively to serve the
needs of other disciplines.
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Some Conclusions
If we fail to offer courses that meet the needs
of the students in the other disciplines, those
departments will increasingly drop the
requirements for math courses. This is already
starting to happen in engineering. Math
departments may well end up offering little
beyond developmental algebra courses that serve
little purpose.
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• Responding to the
• Challenges

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Four Special Invited Conferences

• Rethinking the Preparation for Calculus,
• Forum on Quantitative Literacy,
• CRAFTY Curriculum Foundations Project,
• Reforming College Algebra,

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Common Recommendations

• College Algebra courses should stress
  conceptual understanding, not rote manipulation.
• College Algebra courses should be real-world
  problem based
• Every topic should be introduced through a
  real-world problem and then the mathematics
  necessary to solve the problem is developed.

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Common Recommendations

• College Algebra courses should focus on
  mathematical modelingthat is,
• transforming a real-world problem into
  mathematics using linear, exponential and power
  functions, systems of equations, graphing, or
  difference equations.
• using the model to answer problems in context.
• interpreting the results and changing the model
  if needed.

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Common Recommendations

• College Algebra courses should emphasize
  communication skills reading, writing,
  presenting, and listening.
• These skills are needed on the job and for
  effective citizenship as well as in academia.
• College Algebra courses should make
  appropriate use of technology to enhance
  conceptual understanding, visualization, inquiry,
  as well as for computation.

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Common Recommendations

• College Algebra courses should feature
  student-centered rather than instructor-centered
  pedagogy.
• - They should include hands-on activities rather
  than be all lecture.
• - They should emphasize small group projects
  involving inquiry and inference.

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Important Volumes

• AMATYC Crossroads Standards.
• NCTM, Principles and Standards for School
  Mathematics.
• CUPM Curriculum Guide Undergraduate Programs
  and Courses in the Mathematical Sciences, MAA
  Reports.
• Ganter, Susan and Bill Barker, Eds., A
  Collective Vision Voices of the Partner
  Disciplines, MAA Reports.

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Important Volumes

• Madison, Bernie and Lynn Steen, Eds.,
  Quantitative Literacy Why Numeracy Matters for
  Schools and Colleges, National Council on
  Education and the Disciplines, Princeton.
• Baxter Hastings, Nancy, Flo Gordon, Shelly
  Gordon, and Jack Narayan, Eds., A Fresh Start
  for Collegiate Mathematics Rethinking the
  Courses below Calculus, MAA Notes.

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AMATYC Crossroads Standards

• In general, emphasis on the meaning and use of
  mathematical ideas must increase, and attention
  to rote manipulation must decrease.
• Faculty should include fewer topics but cover
  them in greater depth, with greater
  understanding, and with more flexibility. Such
  an approach will enable students to adapt to new
  situations.
• Areas that should receive increased attention
  include the conceptual understanding of
  mathematical ideas.

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CUPM Curriculum Guide

• All students, those for whom the (introductory
  mathematics) course is terminal and those for
  whom it serves as a springboard, need to learn to
  think effectively, quantitatively and logically.
• Students must learn with understanding, focusing
  on relatively few concepts but treating them in
  depth. Treating ideas in depth includes
  presenting each concept from multiple points of
  view and in progressively more sophisticated
  contexts.

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CUPM Curriculum Guide

• A study of these (disciplinary) reports and the
  textbooks and curricula of courses in other
  disciplines shows that the algorithmic skills
  that are the focus of computational college
  algebra courses are much less important than
  understanding the underlying concepts.
• Students who are preparing to study calculus
  need to develop conceptual understanding as well
  as computational skills.

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• Voices of the Partner Disciplines
• CRAFTYs Curriculum Foundations Project

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Curriculum Foundations Project
A series of 11 workshops with leading educators
from 17 quantitative disciplines to inform the
mathematics community of the current mathematical
needs of each discipline. The results are
summarized in the MAA Reports volume A
Collective Vision Voices from the Partner
Disciplines, edited by Susan Ganter and Bill
Barker.

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What the Physicists Said

• Students need conceptual understanding first,
  and some comfort in using basic skills then a
  deeper approach and more sophisticated skills
  become meaningful. Computational skill without
  theoretical understanding is shallow.

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What the Physicists Said

• Students should be able to focus a situation
  into a problem, translate the problem into a
  mathematical representation, plan a solution, and
  then execute the plan. Finally, students should
  be trained to check a solution for reasonableness.

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What the Physicists Said

• The learning of physics depends less directly
  than one might think on previous learning in
  mathematics. We just want students who can
  think. The ability to actively think is the most
  important thing students need to get from
  mathematics education.

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What Business Faculty Said

• Courses should stress conceptual understanding
  (motivating the math with the whys not just
  the hows).
• Students should be comfortable taking a problem
  and casting it in mathematical terms.
• Courses should use industry standard technology
  (spreadsheets).

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Common Themes from All Disciplines

• Strong emphasis on problem solving
• Strong emphasis on mathematical modeling
• Conceptual understanding is more important than
  skill development
• Development of critical thinking and reasoning
  skills is essential

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Common Themes from All Disciplines

• Use of technology, especially spreadsheets
• Development of communication skills (written and
  oral)
• Greater emphasis on probability and statistics
• Greater cooperation between mathematics and the
  other disciplines

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• A Fresh Start for Collegiate Mathematics
• Rethinking the Courses below Calculus
• Nancy Baxter Hastings
• Florence Gordon
• Sheldon Gordon
• Jack Narayan
• MAA Notes, January 2006

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A Fresh Start to Collegiate Math

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A Fresh Start to Collegiate Math

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A Fresh Start to Collegiate Math

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A Fresh Start to Collegiate Math

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A Fresh Start to Collegiate Math

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A Fresh Start to Collegiate Math

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A Fresh Start to Collegiate Math

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Distribution Plan

• With support from the NSF, the MAA has developed
  a distribution plan to provide one free copy to
  any department that requests one.
• Announcements will be sent to all department
  chairs informing them of the details in February.

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Common Themes
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Common Themes

• Conceptual Understanding is more important than
  rote manipulation
• The Rule of Four Graphical, Numerical,
  Algebraic and Verbal Representations
• Realistic Applications via Math Modeling
• Non-routine problems and assignments
• Algebra in Context Not Just Drill

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Common Themes

• Families of Functions Linear, Exponential,
  Power, Logarithmic, Polynomial, and Sinusoidal
• The significance of the parameters in the
  different families of functions
• Limitations of the models developed
• the practical significance of the domain and
  range

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Common Themes

• Data Analysis
• Connections to Other Disciplines
• Writing and Communication
• More Active Classroom Environment Group Work,
  Collaborative Learning, Exploratory Approach to
  Mathematics
• Use of Technology in Teaching and Learning

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Conceptual Understanding

• What does conceptual understanding mean?
• How do you recognize its presence or absence?
• How do you encourage its development?
• How do you assess whether students have
  developed conceptual understanding?

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What Does the Slope Mean?
Comparison of student response to a problem on
the final exams in Traditional vs. Reform College
Algebra/Trig Brookville College enrolled 2546
students in 1996 and 2702 students in 1998.
Assume that enrollment follows a linear growth
pattern. a. Write a linear equation giving the
enrollment in terms of the year t. b. If the
trend continues, what will the enrollment be in
the year 2016? c. What is the slope of the line
you found in part (a)? d. Explain, using an
English sentence, the meaning of the slope. e.
If the trend continues, when will there be 3500
students?
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Responses in Traditional Class

• 1. The meaning of the slope is the amount that
  is gained in years and students in a given
  amount of time.
• 2. The ratio of students to the number of years.
  
• 3. Difference of the ys over the xs.
• 4. Since it is positive it increases.
• 5. On a graph, for every point you move to the
  right on the x- axis. You move up 78 points on
  the y-axis.
• 6. The slope in this equation means the students
  enrolled in 1996. Y MX B .
• 7. The amount of students that enroll within a
  period of time.
• Every year the enrollment increases by 78
  students.
• The slope here is 78 which means for each unit of
  time, (1 year) there are 78 more students
  enrolled.

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Responses in Traditional Class
10. No response 11. No response 12. No
response 13. No response 14. The change in
the x-coordinates over the change in the
y- coordinates. 15. This is the rise in the
number of students. 16. The slope is the average
amount of years it takes to get 156 more
students enrolled in the school. 17. Its how
many times a year it increases. 18. The slope is
the increase of students per year.
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Responses in Reform Class

• 1. This means that for every year the number of
  students increases by 78.
• 2. The slope means that for every additional
  year the number of students increase by 78.
• 3. For every year that passes, the student
  number enrolled increases 78 on the previous
  year.
• As each year goes by, the of enrolled students
  goes up by 78.
• This means that every year the number of enrolled
  students goes up by 78 students.
• The slope means that the number of students
  enrolled in Brookville college increases by 78.
• Every year after 1996, 78 more students will
  enroll at Brookville college.
• Number of students enrolled increases by 78 each
  year.

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Responses in Reform Class

• 9. This means that for every year, the amount of
  enrolled students increase by 78.
• 10. Student enrollment increases by an average
  of 78 per year.
• 11. For every year that goes by, enrollment
  raises by 78 students.
• 12. That means every year the of students
  enrolled increases by 2,780 students.
• 13. For every year that passes there will be 78
  more students enrolled at Brookville college.
• The slope means that every year, the enrollment
  of students increases by 78 people.
• Brookville college enrolled students increasing
  by 0.06127.
• Every two years that passes the number of
  students which is increasing the enrollment into
  Brookville College is 156.

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Responses in Reform Class
17. This means that the college will enroll
.0128 more students each year. 18. By every
two year increase the amount of students goes up
by 78 students. 19. The number of students
enrolled increases by 78 every 2 years.
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Understanding Slope
Both groups had comparable ability to calculate
the slope of a line. (In both groups, several
students used ?x/?y.)
It is far more important that our students
understand what the slope means in context,
whether that context arises in a math course, or
in courses in other disciplines, or eventually on
the job.
Unless explicit attention is devoted to
emphasizing the conceptual understanding of what
the slope means, the majority of students are not
able to create viable interpretations on their
own. And, without that understanding, they are
likely not able to apply the mathematics to
realistic situations.
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Further Implications

• If students cant make their own connections with
  a concept as simple as slope, they wont be able
  to create meaningful interpretations on their own
  for more sophisticated concepts. For instance,
• What is the significance of the base (growth or
  decay factor) in an exponential function?
• What is the meaning of the power in a power
  function?
• What do the parameters in a realistic sinusoidal
  model tell about the phenomenon being modeled?
• What is the significance of the factors of a
  polynomial?
• What is the significance of the derivative?
• What is the significance of a definite integral?
  

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Further Implications
If we focus only on manipulative skills without
developing conceptual understanding, we produce
nothing more than students who are only Imperfect
Organic Clones of a TI-89
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Results of the Study
The study involved 10 common questions on the
final exam in college algebra/trigonometry, most
of which were basically computational in
nature. The students in the reform sections
outscored those in the traditional,
algebraic-oriented, sections, on 7 of the 10
questions.
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Follow-Up Results in Calculus
The students involved in the precalculus study
were then followed in Calculus I the next
term. The calculus course was a reform course
with emphasis also on conceptual understanding,
not just manipulation.
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Follow-Up Results in Calculus
On every weekly quiz, on every class test, and on
the final exam, the students from the reform
sections of precalculus consistently scored
higher than the students from the traditional
sections. On an attitudinal survey, the students
from the reform section had significantly better
attitudes toward mathematics, its usefulness, and
the importance of technology for problem solving.
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Follow-Up Results in Calculus
77 of the students who had been in a reform
section of precalculus ended up receiving a
passing grade in Calculus I. 41 of those who
had been in a traditional section of precalculus
received a passing grade in Calculus I.
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Developing Conceptual Understanding
Conceptual understanding cannot be just an
add-on. It must permeate every course and be a
major focus of the course. Conceptual
understanding must be accompanied by realistic
problems in the sense of mathematical
modeling. Conceptual problems must appear in all
sets of examples, on all homework assignments, on
all project assignments, and most importantly, on
all tests. Otherwise, students will not see them
as important.
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Conclusions
We cannot simply concentrate on teaching the
mathematical techniques that the students need.
It is as least as important to stress conceptual
understanding and the meaning of the mathematics.
We can accomplish this by using a combination
of realistic and conceptual examples, homework
problems, and test problems that force students
to think and explain, not just manipulate
symbols. If we fail to do this, we are not
adequately preparing our students for successive
mathematics courses, for courses in other
disciplines, and for using mathematics on the job
and throughout their lives.
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• Some Illustrative Examples
• of Problems
• to Develop or Test for
• Conceptual Understanding

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Identify each of the following functions (a) -
(n) as linear, exponential, logarithmic, or
power. In each case, explain your
reasoning.(g) y 1.05x (h) y x1.05
(i) y (0.7)t (j) y v0.7
(k) z L(-½) (l) 3U 5V 14
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For the polynomial shown,(a) What is the
minimum degree? Give two different reasons for
your answer.(b) What is the sign of the leading
term? Explain.(c) What are the real roots?(d)
What are the linear factors? (e) How many
complex roots does the polynomial have?
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The following table shows world-wide wind power
generating capacity, in megawatts, in various
years.
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(a) Which variable is the independent variable
and which is the dependent variable? (b) Explain
why an exponential function is the best model to
use for this data. (c) Find the exponential
function that models the relation-ship between
power P generated by wind and the year t. (d)
What are some reasonable values that you can use
for the domain and range of this function? (e)
What is the practical significance of the base in
the exponential function you created in part
(c)? (f) What is the doubling time for this
exponential function? Explain what it means.
(g) According to your model, what do you predict
for the total wind power generating capacity in
2010?
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Biologists have long observed that the larger the
area of a region, the more species live there.
The relationship is best modeled by a power
function. Puerto Rico has 40 species of
amphibians and reptiles on 3459 square miles and
Hispaniola (Haiti and the Dominican Republic) has
84 species on 29,418 square miles. (a)
Determine a power function that relates the
number of species of reptiles and amphibians on a
Caribbean island to its area. (b) Use the
relationship to predict the number of species of
reptiles and amphibians on Cuba, which measures
44218 square miles.
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The accompanying table and associated scatterplot
give some data on the area (in square miles) of
various Caribbean islands and estimates on the
number species of amphibians and reptiles living
on each.
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(a) Which variable is the independent variable
and which is the dependent variable? (b) The
overall pattern in the data suggests either a
power function with a positive power p lt 1 or a
logarithmic function, both of which are
increasing and concave down. Explain why a power
function is the better model to use for this
data. (c) Find the power function that models
the relationship between the number of species,
N, living on one of these islands and the area,
A, of the island and find the correlation
coefficient. (d) What are some reasonable
values that you can use for the domain and range
of this function? (e) The area of Barbados is 166
square miles. Estimate the number of species of
amphibians and reptiles living there.
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Write a possible formula for each of the
following trigonometric functions
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The average daytime high temperature in New York
as a function of the day of the year varies
between 32?F and 94?F. Assume the coldest day
occurs on the 30th day and the hottest day on the
214th. (a) Sketch the graph of the temperature
as a function of time over a three year time
span. (b) Write a formula for a sinusoidal
function that models the temperature over the
course of a year. (c) What are the domain and
range for this function? (d) What are the
amplitude, vertical shift, period, frequency, and
phase shift of this function? (e) Predict the
high temperature on March 15. (f) What are all
the dates on which the high temperature is most
likely 80??
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Some Other IssuesRegarding the Needto Refocus
the Coursesbelow Calculus
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The Need to RethinkPlacementin Mathematics
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Rethinking Placement Tests

• Four scenarios
• Students come from traditional curriculum into
  traditional curriculum.
• Students from Standards-based curriculum into
  traditional curriculum.
• Students from traditional curriculum into reform
  curriculum.
• 4. Students from Standards-based curriculum into
  reform curriculum.

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A National Placement Test
1. Square a binomial. 2. Determine a
quadratic function arising from a verbal
description (e.g., area of a rectangle whose
sides are both linear expressions in x). 3.
Simplify a rational expression. 4. Confirm
solutions to a quadratic function in factored
form. 5. Completely factor a polynomial. 6.
Solve a literal equation for a given unknown.

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A National Placement Test
7. Solve a verbal problem involving
percent. 8. Simplify and combine like
radicals. 9. Simplify a complex fraction. 10.
Confirm the solution to two simultaneous linear
equations. 11. Traditional verbal problem
(e.g., age problem). 12. Graphs of linear
inequalities.

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A Modern High School Problem
Given the complete 32-year set of monthly CO2
emission levels (a portion is shown below),
create a mathematical model to fit the data.

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A Modern High School Problem
1. Students first do a vertical shift of about
300 ppm and then fit an exponential function to
the transformed data to get

2. They then create a sinusoidal model to fit
the monthly oscillatory behavior about the
exponential curve
3. They then combine the two components to get
4. They finally give interpretations of the
various parameters and what each says about the
increase in concentration and use the model to
predict future or past concentration levels.
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Placement, Revisited
Picture an entering freshman who has taken high
school courses with a focus on problems like the
preceding one and who has developed an
appreciation for the power of mathematics based
on understanding the concepts and applying them
to realistic situations. What happens when that
student sits down to take a traditional placement
test? Is it surprising that many such students
end up being placed into developmental courses?

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What a High School Teacher Said

• If you try to teach my students with the
  mistaken belief that they know the mathematics I
  knew at their age, you will miss a great
  opportunity. My students know more mathematics
  than I did, but it is not the same mathematics
  and I believe they know it differently. They
  have a different vision of mathematics that would
  be helpful in learning calculus if it were
  tapped.
• Dan Teague

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The Need to RethinkCourse Content
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What Can Be Removed?
How many of you remember that there used to be
something called the Law of Tangents? What
happened to this universal law? Did triangles
stop obeying it? Does anyone miss it?
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What Can Be Removed?

• Descartes rule of signs
• The rational root theorem
• Synthetic division
• The Cotangent, Secant, and Cosecant
• were needed for computational purposes.
• Just learn and teach a new identity

87
How Important Are Rational Functions?

• In DE To find closed-form solutions for
  several differential equations, (usually done
  with CAS today, if at all)
• In Calculus II Integration using partial
  fractionsoften all four exhaustive (and
  exhausting) cases
• In Calculus I Differentiating rational
  functions
• In Precalculus Emphasis on the behavior of
  all kinds of rational functions and even
  partial fraction decompositions
• In College Algebra Addition, subtraction,
  multiplication, division and especially
  reduction of complex fractional expressions
• In each course, it is the topic that separates
  the adults from the children! But, can you
  name any realistic applications that involve
  rational functions? Why do we need them in
  excess?

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Challenges to Be Faced
89
The Challenges Ahead

• Convincing the math community
• 1. Conducting a series of extensive tracking
  studies to determine how many (or how few)
  students who take these courses actually go on to
  calculus.
• 2. Identifying and highlighting best practices
  in programs that reflect the goals of this
  initiative.

90
The Challenges Ahead

• Convincing college administrators to support
  (both academically and financially) efforts to
  refocus the courses below calculus.

91
What Can Administrators Do?
When the University of Michigan wanted to change
to calculus reform, including going from large
lectures of 800 students to small classes of 20
taught by full-time faculty, the department
argued to the dean that by saving only 2 of the
students who fail out because of calculus, the
savings to the university would exceed the
1,000,000 annual additional instructional cost.
The dean immediately said Go for it.
92
The Challenges Ahead
Convincing academic bodies outside of
mathematics to allow alternatives to traditional
college algebra courses to fulfill general
education requirements.
93
An Example Georgia
The state education department in Georgia had a
mandate for general education that every student
must take college algebra. A group of faculty
from various two and four year colleges across
the state lobbied for years until they finally
convinced the state authorities to allow a course
in mathematical modeling at the college algebra
level to serve as an alternative for satisfying
the Gen Ed math requirement.
94
The Challenges Ahead
Convincing the testing industry to begin
development of a new generation of placement and
related tests that reflect the NCTM
Standards-based curricula in the schools and the
kinds of refocused courses below calculus in the
colleges that we hope to being about.
95
The Challenges Ahead

• Gaining the active support of representatives of
  a wide variety of other disciplines in the effort
  to refocus the courses below calculus.
• CRAFTY and MAD (Math Across the Disciplines)
  committee have launched a second round of
  Curriculum Foundations workshops to address this
  issue.

96
The Challenges Ahead

• Gaining the active support of representatives of
  business, industry, and government in this
  initiative.
• Discussions are underway about revisiting some
  of the participants in the Forum on Quantitative
  Literacy.

97
The Challenges Ahead

• Developing a faculty development program to
  assist faculty, especially part time faculty and
  graduate TAs, to teach the new versions of these
  courses.
• NSF has funded a demonstration project through
  CRAFTY involving 11 schools.
• 200 other departments wanted to be part of this
  project.

98
The Challenges Ahead
Influencing teacher preparation programs to
rethink the courses they offer to prepare the
next generation of teachers in the spirit of this
initiative.
99
The Challenges Ahead
Influencing funding agencies such as the NSF to
develop new programs that are specifically
designed to promote both the development of new
approaches to the courses below calculus and the
widespread implementation of existing reform
versions of these courses.