The Mathematics Curriculum

Mathematics for the Laboratory Sciences:

College Algebra, Precalculus,and Up

Sheldon P. [email protected]

The Mathematics CurriculumAt most schools, the mathematics curriculum is focused

on moving students up the mathematics pipeline:

either to become math majors or to serve the traditional needs of engineering and physics curricula.

But these students are only a small minority of the students we see and whose needs we should be serving.

Bachelor’s Degrees in Mathematics

In 2005,

P There were 1,439,264 bachelor’s degrees

P Of these, 14,351 were in mathematics

This is less than one percent!

(There are 23,000 degrees in recreation and leisure!)

The Needs of Our Students

The reality is that virtually none of the students

we face are going to be math majors.

They take our courses because of requirements

from other disciplines.

What do those other disciplines want their

students to bring from math courses?

Voices of the Partner Disciplines

CRAFTY’s Curriculum Foundations Project

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 of the Partner Disciplines, edited by Susan Ganter and Bill Barker.

What the Physicists Said

• Conceptual understanding of basic mathematical principles is very important for success in introductory physics. It is more important than esoteric computational skill. However, basic computational skill is crucial.

• Development of problem solving skills is a critical aspect of a mathematics education.


• Courses should cover fewer topics and place increased emphasis on increasing the confidence and competence that students have with the most fundamental topics.


• 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.


• 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.


• 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.

What the Biologists Said

• New areas of biological investigation have resulted in an increase in quantification of biological theories and models.

• The collection and analysis of data that is central to biology inevitably leads to the use of mathematics.

• Mathematics provides a language for the development and expression of biological concepts and theories. It allows biologists to summarize data, to describe it in logical terms, to draw inferences, and to make predictions.


• Statistics, modeling and graphical representation should take priority over calculus.

• The teaching of mathematics and statistics should use motivating examples that draw on problems or data taken from biology.

• Creating and analyzing computer simulations of biological systems provides a link between biological understanding and mathematical theory.

What the Biologists SaidThe quantitative skills needed for biology:• The meaning and use of variables, parameters, functions,

and relations.• To formulate linear, exponential, and logarithmic

functions from data or from general principles.• To understand the periodic nature of the sine and cosine

functions.• The graphical representation of data in a variety of

formats – histograms, scatterplots, log-log graphs (for power functions), and semi-log graphs (for exponential and log functions).


Other quantitative skills:• Some calculus for calculating areas and average

values, rates of change, optimization, and gradients for understanding contour maps.

• Statistics – descriptive statistics, regression analysis, multivariate analysis, probability distributions, simulations, significance and error analysis.

• Discrete Mathematics and Matrix Algebra – graphs (trees, networks, flowcharts, digraphs), matrices, and difference equations.


• The sciences are increasingly seeing students who are quantitatively ill-prepared.• The biological sciences represent the largest science client of mathematics education.• The current mathematics curriculum for biology majors does not provide biology students with appropriate quantitative skills. • The biologists suggested the creation of mathematics courses designed specifically for biology majors.• This would serve as a catalyst for needed changes in the undergraduate biology curriculum.• We also have to provide opportunities for the biology faculty to increase their own facility with mathematics.

What the Chemists Said

• Six themes for the mathematical preparation of chemistry students emerged. Mathematicians are asked to keep these six themes in mind as courses in mathematics are redesigned.

What the Chemists Said

• Multivariate Relationships: Almost all problems in chemistry from the lowly ideal gas law to the most sophisticated applications of quantum mechanics and statistical mechanics are multivariate.

• Numerical Methods: Used in a host of practical calculations, most enabled by the use of computers

What the Chemists Said• Visualization: Chemistry is highly visual. Practitioners

need to visualize structures and atomic and molecular orbitals in three dimensions.

• Scale and Estimation: The stretch from the world of atoms and molecules to tangible materials is of the order of Avogadro’s number, about 1024. Distinctions of scale, along with an intuitive feeling for the different values along the scales of size, are of central importance in chemistry.

What the Chemists Said• Mathematical Reasoning: Facility at mathematical

reasoning permeates most of chemistry. Students must be able to follow algebraic arguments if they are to understand the relationships between mathematical expressions, to adapt these expressions to applications, and to see that most specific mathematical expressions can be recovered from a few fundamental relationships in a few steps.

What the Chemists Said• Data Analysis: Data analysis is a widespread

activity in chemistry that depends on the application of mathematical methods. These methods include statistics and curve fitting.

Health-Related Life Sciences• “Many participants put special emphasis on the use of

models.” “Models are a way of organizing information for the purpose of gaining insight and providing intuition into systems that are too complex to understand any other way”.

• “Students should master a higher level interface, e.g.: spreadsheet, symbolic/numerical computational packages( e.g. Mathematica, Maple, Matlab), statistical packages.

• BE FLEXIBLE: package topics creatively thru long-term interaction between mathematics and the life sciences.

What Business Faculty Said

• Courses should stress problem solving, with the incumbent recognition of ambiguities.• Courses should stress conceptual understanding (motivating the math with the “why’s” – not just the “how’s”).• Courses should stress critical thinking.• An important student outcome is their ability to develop appropriate models to solve defined problems.


Mathematics is an integral component of the business school curriculum. Mathematics Departments can help by stressing conceptual understanding of quantitative reasoning and enhancing critical thinking skills. Business students must be able not only to apply appropriate abstract models to specific problems but also to become familiar and comfortable with the language of and the application of mathematical reasoning. Business students need to understand that many quantitative problems are more likely to deal with ambiguities than with certainty. In the spirit that less is more, coverage is less critical than comprehension and application.


• Courses should use industry standard technology (spreadsheets).

• An important student outcome is their ability to become conversant with mathematics as a language. Business faculty would like its students to be comfortable taking a problem and casting it in mathematical terms.

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 and on the job

• Fitting functions to data

• Statistical reasoning

• Recursion and difference equations – the

mathematical language of spreadsheets

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?

Everybody talks about emphasizing Conceptual Understanding, but

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?

Responses in Reform Class1. 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.4. As each year goes by, the # of enrolled students goes up by 78.5. This means that every year the number of enrolled students

goes up by 78 students.6. The slope means that the number of students enrolled in

Brookville college increases by 78.7. Every year after 1996, 78 more students will enroll at

Brookville college.8. Number of students enrolled increases by 78 each year.

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.14. The slope means that every year, the enrollment of students

increases by 78 people.15. Brookville college enrolled students increasing by 0.06127.16. Every two years that passes the number of students which is

increasing the enrollment into Brookville College is 156.

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.

Responses in Traditional Class1. 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 y’s over the x’s.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.8. Every year the enrollment increases by 78 students.9. The slope here is 78 which means for each unit of time, (1

year) there are 78 more students enrolled.

Responses in Traditional Class

10. No response11. No response12. 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.

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.

Further Implications

If students can’t make their own connections with a concept as simple as the slope of a line, they won’t be able to create meaningful interpretations and connections on their own for more sophisticated mathematical 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 of a function? • What is the significance of a definite integral?

Further ImplicationsIf we focus only on developing

manipulative skills

without developing

conceptual understanding,

we produce nothing more than students

who are only

Imperfect Organic Clones

of a TI-89

Should x Mark the Spot?All other disciplines focus globally on the entire universe of a through z, with the occasional contribution of through .

Only mathematics focuses on a single spot, called x.

Newton’s Second Law of Motion: y = mx,

Einstein’s formula relating energy and mass: y = c2x,

The Ideal Gas Law: yz = nRx.

Students who see only x’s and y’s do not make the connections and cannot apply the techniques when other letters arise in other disciplines.

Should x Mark the Spot?

Kepler’s third law expresses the relationship between the

average distance of a planet from the sun and the length

of its year.

If it is written as y2 = 0.1664x3, there is no suggestion of

which variable represents which quantity.

If it is written as t2 = 0.1664D3 , a huge conceptual

hurdle for the students is eliminated.

Should x Mark the Spot?

When students see 50 exercises

where the first 40 involve solving for x,

and a handful at the end involve other letters,

the overriding impression they gain is that x is the only

legitimate variable and the few remaining cases are just

there to torment them.

Some Illustrative Examples and Problems

for Conceptual Understanding

and Mathematical Modeling

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)x (j) y = x0.7

(k) y = x(-½) (l) 3x - 5y = 14

(m) x y (n) x y

0 3 0

5

1 5.1 1

7

2 7.2 2

9.8

3 9.3 3

13.7

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?

The following table shows world-wide wind power generating capacity, in megawatts, in various

years.

Year 1980 1985 1988 1990 1992 1995 1997 1999

Windpower 10 1020 1580 1930 2510 4820 7640 13,840

0

5000

10000

15000

1980 1985 1990 1995 2000

(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 relationship 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 does it means. (g) According to your model, what do you predict for the total wind power generating capacity in 2010?

A Temperature Experiment

An experiment is conducted to study the rate at which temperature changes. A temperature probe is first heated in a cup of hot water and then pulled out and placed into a cup of cold water. The temperature of the probe, in PC, is measured every second for 36 seconds and recorded in the following table.

Time 1 2 3 4 5 6 7 842.3 36.03 30.85 26.77 23.58 20.93 18.79 17.08

31 32 33 34 35 368.78 8.78 8.78 8.78 8.66 8.66

Find a function that fits this data.

A Temperature Experiment

5

10

15

20

25

30

35

40

45

Tem

pera

ture

(degre

es C

)

time (1 - 36 seconds)

The data suggest an exponential decay function,but the data do not decay to 0. To find a function, one first has to shift the data values down to get a transformed set of data that decay to 0. y = T – 8.6 = 35.439(0.848)t

Then one has to fit an exponential function to the transformed data. Finally, one has to undo the transformation by shifting the resulting exponential function. T = 8.6 + 35.439(0.848)t.

Modeling the Decay of a DrugEvery drug is washed out of the bloodstream, usually by the kidneys, but at different rates.

For example, in any 24-hour period, about 25% of any Prozac in the blood is washed out, leaving 75% of the amount. This suggests an exponential decay function. If the initial dosage is 80 mg. then the model is D(t) = 80 (0.75)t, t in days.

Predictive Questions

What will the level of Prozac (or any drug) be after 7 days (or any given number of time periods)?

How long will it take until the level of Prozac is down to 10 mg (or to any given level)?

What is the half-life of Prozac in the blood?

Repeated Doses of a Drug

• 25% of the Prozac in the blood is washed out each day, leaving 75%

• Typical dose is 40 mg each day

Level of Prozac

D0 = 40

D1 = .75(40) + 40 = 30 + 40 = 70

D2 = .75(70) + 40 = 92.5

D3 = .75(92.5) + 40 = 109.375

{40, 70, 92.5, 109.375, 122.031, 131.523, …}

Level of Prozac D0 = 40

D1 = .75 D0 + 40

D2 = .75(D1 ) + 40

D3 = .75(D2) + 40

D4 = .75(D3) + 40

In general, after any number of days n, we have the difference equation:

Dn+1 = .75 Dn + 40

Difference Equation Model

Dn+1 = .75 Dn + 40

D0 = 40

D1 = .75(D0 ) + 40 = 70

D2 = .75(D1 ) + 40 = 92.5

D3 = .75(D2 ) + 40 = 109.375

D4 = .75(D3 ) + 40 = 122.031

Solution to the Difference Equation:

{40, 70, 92.5, 109.375, 122.031, 131.523, …}

Solution of the Difference Equation

0

40

80

120

160

200

0 5 10 15 20Number of Days

Lev

el o

f P

roza

c

What Happens if an Overdose?

D0 = 400

D1 = .75(400) + 40 = 340

D2 = .75(340) + 40 = 295

D3 = .75(295) + 40 = 261.25

0

40

80

120

160

200240

280

320

360

400

0 5 10 15 20Number of Days

Level

of

Pro

zac

Finding the Maintenance Level

Dn+1 = 0.75 Dn + 40

Assume that Dn+1 = L and Dn = L

L = 0.75L + 40

L = 40 /0.25 = 160 mg.

Creating a Formula for the Solution

n 0 1 2 3 4 5

Dn 40 70 92.5 109.4 122.0 131.5

160-Dn 120 90 67.5 50.6 38.0 28.5

160 - Dn = 119.99961 (0.7500021) n

Dn = 160 - 119.99961 (0.7500021) n

How Well Does it Fit?

0

50

100

150

200

0 5 10 15 20

Number of days

160-

Dn

Finding the Solution in General

Diff. Eqn: Dn+1 = 0.75 Dn + 40

Solution: Dn = 160 - 120(0.75) n

In general:

Dn+1 = b Dn + C

Dn = L - (L - D0) bn

Proving the Solution in General

Dn+1 = b Dn + C

Dn = L - (L - D0) bn

LHS: Dn+1 = L - (L - D0) bn+1

RHS: b Dn + C = bL - b(L - D0) bn + C

= bL - (L - D0) bn+1 + C

= bL - (L - D0) bn+1 + C

= bL - (L - D0) bn+1 + L(1 - b)

= L - (L - D0) bn+1

The Species-Area ModelBiologists 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.

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.

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

0

20

40

60

80

100

0 15000 30000 45000

Area (square miles)

Num

ber o

f Spe

cies

(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 Dead Body Problem

Mr. Jones' body was found in his kitchen at 9 am by the police who noted that the body temperature was 77.3P F and that the room temperature was 70P. An hour later, the medical examiner found the body temperature was 76.1P. You may presume that Mr. Jones' body temperature was the normal reading of 98.6P at the time of death. At what time was he murdered?

Write a possible formula for each of the following trigonometric functions:

The average daytime high temperature in New York as a function of the day of the year varies between 32F and 94F. 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 most likely high temperature on March 15.(f) What are all the dates on which the high temperature is most likely 80?

Balancing Chemical ReactionsPhotosynthesis

Carbon dioxide (CO2) plus Water (H2O) produces Glucose (C6H12O6) plus Oxygen (O2).

How many molecules of each are needed?

If Glucose is the “target” molecule, then

x CO2 + y H2O - z O2 = 1 C6H12O6

Carbon: 1x + 0y – 0z = 6Oxygen: 2x + 1y - 2z = 6Hydrogen: 0x + 12y - 0z = 2

Need to solve this linear system of 3 equations in 3 unknowns

Farmingdale’s Math & Bio Project

Almost all math and bio curriculum 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.

The Mathematical Needs of Biology

• In discussions with biology faculty, it became clear that most courses for non-majors (and even those for majors in some areas) use almost no mathematics in class.

• Mathematics arises almost exclusively in the lab when students have to analyze experimental data and then their weak math skills become dramatically evident.

Farmingdale’s Math & Bio Project

Our original plan was to develop 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.

Farmingdale Math & Bio 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 TopicsWeek 1 Behavior of Functions Intro to Measurements

and Measuring

Week 2 Families of Functions,

Linear Functions

Linear Growth – part 1

Week 3 Linear Functions

and Linear Regression

Linear Growth – part 2

Week 4 Exponential Growth

and Decay Functions

Exponential Growth

Week 5 Exponential Regression

and Power Functions

Exponential Decay

part 1

Week 6 Power Functions and

Polynomials

Exponential Decay –

part 2

Course TopicsWeek 7 New functions from Old Power Function growthWeek 8 Logistic and Surge

Functions

Logarithmic Functions

Week 9 Matrix Models and

Linear Systems

Logistic Growth

Week 10 Sinusoidal Functions and Periodic Behavior

Surge Functions

Week 11 Periodic Functions – part 2

Polynomial growth

Week 12 Probability Models Periodic BehaviorWeek 13 Probability Models and

Difference Equations

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 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.


The math department has created a new four credit precalculus course to serve the needs of the biology students – basically, the same math course without the lab.The focus is on conceptual understanding, data analysis, statistical reasoning, and mathematical modeling, not on developing algebraic skills (other than in a few special cases where algebra is needed to solve problems that arise naturally in context).


The math department has also created a new two-semester calculus sequence for biology students – it also emphasizes concepts over manipulation and stresses biological applications.The math department has created a one-semester post-precalculus course on mathematical modeling in the biological sciences for bioscience majors and applied math majors.

The Next Challenge

Based on the Curriculum Foundations reports and from discussions with faculty in the lab sciences (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?

The Curriculum Problems We Face

• Students don’t see traditional precalculus or college algebra courses 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.

Integrating Statistics into Mathematics

• Students see the equation of a line in pre-algebra, in elementary algebra, in intermediate algebra, in college algebra, and in precalculus. Yet many still have trouble with it in calculus.• They see statistics ONCE in an introductory statistics course. But statistics is far more complex, far more varied, and often highly counter-intuitive, yet they are then expected to use a wide variety of the statistical ideas and methods in their lab science courses.

Integrating Statistics in Precalculus

Data is Everywhere! We should capitalize on it.

1. A frequency distribution is a function – it can be an effective way 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 that links functions, the real world, and the other disciplines.


3. The normal distribution function is

2 2( ) / 21

2( ) xN x e

It makes for an excellent example involving both stretching and shifting functions and a function of a function.

Integrating Statistics in Precalculus 4. The z-value associated with a measurement x is a nice application of a linear function of x:

xz


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

produces a stretch or a squeeze, depending on the sample size n.

xn

Documents

The Mathematics Curriculum