Transcript Rethinking the Preparation for Calculus October 2001 Washington, DC

Mathematics for the
Biosciences at
Farmingdale State
Sheldon Gordon
[email protected]
Major Premise
“Biology will do for mathematics in
the twenty-first century what physics
did for mathematics in the twentieth
century”
Major Premise
Almost all math and bio projects start at
the calculus level or above.
But the overwhelming majority of
beginning biology students, both majors
and especially non-majors, typically are at
the college algebra or precalculus level.
Most of these students have avoided math
as much as possible.
Our Project
Our original plan was to begin developing
the first stages of a new mathematics
curriculum to serve the needs of biology
students, both the bioscience majors and
the non-majors who take introductory
biology courses.
This would also impact the level of
quantitative work in the biology courses.
Our Project
The project is a collaborative effort
between Farmingdale State College and
Suffolk Community College.
Farmingdale State brings an exceptional
history of successful efforts in reforming
the mathematics curriculum.
Suffolk brings an outstanding record of
utilizing technology and quantitative
methods in its introduction biology courses
and labs.
The Mathematical Needs of Biology
• In discussions with the biology faculty at
both schools, it became clear that most
courses for non-majors (and even those for
majors in some areas) make almost no use of
mathematics in class.
• Math arises almost exclusively in the lab
when students have to analyze data. This is
where their weak math skills are
dramatically evident.
The Farmingdale State Project
Our first step was to develop an alternative
to our modeling-based precalculus course
that would focus almost exclusively on
biological applications.
The course would feature a lab component
taught by the biology faculty, so that each
week’s primary math topic would be
accompanied by an experiment requiring
the use of that mathematical method.
Course Topics
Week 1 Behavior of Functions
Week 2 Families of Functions,
Linear Functions
Week 3 Linear Functions
and Linear Regression
Week 4 Exponential Growth
and Decay Functions
Week 5 Exponential Regression
and Power Functions
Week 6 Power Functions and
Polynomials
Intro to Measurements
and Measuring
Linear Growth – part 1
Linear Growth – part 2
Exponential Growth
Exponential Decay
part 1
Exponential Decay –
part 2
Course Topics
Week 7
Week 8
Week 9
Week 10
Week 11
Week 12
Week 13
New functions from Old
Logistic and Surge
Functions
Matrix Models and
Linear Systems
Sinusoidal Functions
and Periodic Behavior
Periodic Functions –
part 2
Probability Models
Probability Models and
Difference Equations
Power Function growth
Logarithmic Functions
Logistic Growth
Surge Functions
Polynomial growth
Periodic Behavior
Probability Model
(genetics)
What Happened Next
To accommodate the lab component, we
had to change the precalculus course from
four to five credits.
Because of that and conflicts with other
courses (intro chemistry), the biology
students did not register for the course and
it did not run.
What We’ve Done Instead
All the labs in the introductory biology
course are in the process of being changed
to dramatically increase the level of
quantitative experience – the new labs will
incorporate most of the experiments that
were to be part of the precalculus course.
What We’ve Done Instead
The math department has created a new
four credit precalculus course to serve the
needs of the biology students – the same
math course, but no lab.
The focus is on conceptual understanding,
data analysis and mathematical modeling,
rather than on algebraic manipulation
(other than in a few special cases where
needed to solve problems that arise
naturally in context).
What We’ve Done Instead
The math department has also created a
new two-semester calculus sequence for
biology students – it will emphasize
concepts over manipulation and will stress
biological applications.
The math department has created a onesemester post-precalculus mathematical
modeling in the biological sciences course
for bioscience majors and applied math
majors.
Some Sample Problems
Identify each of the following functions (a) - (n) as linear,
exponential, logarithmic, or power. In each case, explain
your reasoning.
(m)
(g) y = 1.05x
(h) y = x1.05
x
y
(n)
x
y
0
0
3
5
1
(i) y =
(0.7)t
(j) y =
v0.7
1
(l) 3U – 5V = 14
7
2
2
(k) z = L(-½)
5.1
7.2
9.8
3
3
9.3
13.7
Some Sample Problems
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 44,218
square miles.
Island
Area
N
Redonda
1
3
Saba
4
5
Montserrat
40
9
Puerto Rico
3459
40
Jamaica
4411
39
Hispaniola
29418
84
Cuba
44218
76
Number of Species
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.
100
80
60
40
20
0
0
15000
30000
Area (square miles)
45000
(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 < 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.
The ocean temperature near New York as a function of
the day of the year varies between 36F and 74F.
Assume the water is coldest on the 40th day and
warmest on the 224th.
(a) Sketch the graph of the water 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) Estimate the water temperature on March 15.
(f) What are all the dates on which the water
temperature is most likely 60?
The Next Challenge
Based on the Curriculum Foundations
reports and from discussions with
faculty in biology (and most other
areas), the most critical mathematical
need of the other disciplines is for
students to know more about statistics.
How do we integrate statistical ideas
and methods into math courses at all
levels?
What Students Really Need from Math
The ability to make sense of data – to
interpret graphs and tables
•
• Statistical measures of data
• Estimating the mean of a population
from a sample
The Curriculum Problems We Face
• Students don’t see college algebra or precalculus
as providing any useful skills for their other
courses.
• Typically, college algebra is the prerequisite for
introductory statistics.
• Introductory statistics is already overly
crammed with too much information.
• Most students put off taking the math as long as
possible. So most don’t know any of the statistics
when they take the courses in bio or other fields.
Some Ideas for College Algebra
Data is Everywhere! We should capitalize on it.
1. A frequency distribution is a function – it is an
effective way even to introduce and develop the
concept of function.
2. Data analysis – the idea of fitting linear,
exponential, power, polynomial, sinusoidal and
other functions to data – is already becoming a
major theme in some college algebra courses. It
can be the unifying theme to link functions, the
real world, and the other disciplines.
Some Ideas for College Algebra
3. The normal distribution function is
N ( x) 
1
 2
e
 ( x   )2 / 2 2
It makes for an excellent example involving
both stretching and shifting functions and
a function of a function.
Some Ideas for College Algebra
4. The z-value associated with a measurement x
is a nice application of a linear function of x:
z
x

Some Ideas for College Algebra
5. The Central Limit Theorem is another
example of stretching and shifting functions -the mean of the distribution of sample means
is a shift and its standard deviation,
x 

n
produces a stretch or a squeeze, depending on
the sample size n.
Some Conclusions
Mathematics is a service department at
almost all institutions. Few, if any, math
departments can exist based solely on
offerings for math and related majors.
And college algebra and related courses
exist almost exclusively to serve the needs
of other disciplines.
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.
Most math departments may well end up
offering little beyond developmental
algebra courses that serve little purpose.
Contact Information
Sheldon P. Gordon
Farmingdale .edu / ~gordonsp
[email protected]