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Refocusing the Courses Below Calculus A Joint

Initiative of MAA, AMATYC NCTM

- This slideshow presentation was created by
- Sheldon P. Gordon
- Farmingdale State University of New York
- gordonsp_at_farmingdale.edu
- with contributions from
- Nancy Baxter Hastings (Dickinson College)
- Florence S. Gordon (NYIT)
- Bernard Madison (University of Arkansas)
- Bill Haver (Virginia Commonwealth University)
- Bill Bauldry (Appalachian State University)
- Permission is hereby granted to anyone to use any

or all of these slides in any related

presentations. - We gratefully acknowledge the support provided

for the development of this presentation package

by the National Science Foundation under grants

DUE-0089400, DUE-0310123, and DUE-0442160. - The views expressed are those of the author and

do not necessarily reflect the views of the

Foundation.

College Algebra and Precalculus

Each year, more than 1,000,000 students take

college algebra and precalculus courses. The

focus in most of these courses is on preparing

the students for calculus. We know that only a

relatively small percentage of these students

ever go on to start calculus.

Some Questions

How many of these students actually ever do go on

to start calculus? How well do the ones who do

go on actually do in calculus?

Some Questions

Why do the majority of these 1,000,000 students

a year take college algebra courses? Are these

students well-served by the kind of courses

typically given as college algebra? If not,

what kind of mathematics do these students really

need?

Enrollment Flows

- Based on several studies of enrollment flows from

college algebra to calculus - Less than 5 of the students who start

college algebra courses ever start Calculus I - The typical DFW rate in college algebra is

typically well above 50 - 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

Some Interesting Studies

In a study at eight public and private

universities in Illinois, Herriott and Dunbar

found that, typically, only about 10-15 of the

students enrolled in college algebra courses had

any intention of majoring in a mathematically

intensive field. At a large two year college,

Agras found that only 15 of the students taking

college algebra planned to major in

mathematically intensive fields.

Some Interesting Studies

- Steve Dunbar has tracked over 150,000 students

taking mathematics at the University of Nebraska

Lincoln for more than 15 years. He found that - only about 10 of the students who pass college

algebra ever go on to start Calculus I - virtually none of the students who pass college

algebra ever go on to start Calculus III. - about 30 of the students who pass college

algebra eventually start business calculus. - about 30-40 of the students who pass

precalculus ever go on to start Calculus I.

Some Interesting Studies

- William Waller at the University of Houston

Downtown tracked the students from college

algebra in Fall 2000. Of the 1018 students who

started college algebra - only 39, or 3.8, ever went on to start Calculus

I at any time over the following three years. - 551, or 54.1, passed college algebra with a C

or better that semester - of the 551 students who passed college algebra,

153 had previously failed college algebra (D/F/W)

and were taking it for the second, third, fourth

or more time

Some Interesting Studies

- The Fall, 2001 cohort in college algebra at the

University of Houston Downtown was slightly

larger. Of the 1028 students who started college

algebra - only 2.8, ever went on to start Calculus I at

any time over the following three years.

The San Antonio Project

The mayors Economic Development Council of San

Antonio recently identified college algebra as

one of the major impediments to the city

developing the kind of technologically

sophisticated workforce it needs. The mayor

appointed a special task force with

representatives from all 11 colleges in the city

plus business, industry and government to change

the focus of college algebra to make the courses

more responsive to the needs of the city, the

students, and local industry.

Why Students Take These Courses

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

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 in

todays technological society

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.

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.

Four Special Invited Conferences

- Rethinking the Preparation for Calculus,
- October 2001.
- Forum on Quantitative Literacy,
- November 2001.
- CRAFTY Curriculum Foundations Project,
- December 2001.
- Reforming College Algebra,
- February 2002.

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.

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.

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.

Common Recommendations

- College Algebra courses should be

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.

Important Volumes

- CUPM Curriculum Guide Undergraduate Programs

and Courses in the Mathematical Sciences, MAA

Reports. - AMATYC Crossroads Standards and the Beyond

Crossroads report. - NCTM, Principles and Standards for School

Mathematics. - Ganter, Susan and Bill Barker, Eds.,
- A Collective Vision Voices of the Partner

Disciplines, MAA Reports.

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.

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.

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.

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.

NCTM Standards

- These recommendations are clearly very much in

the same spirit as the recommendations in NCTMs

Principles and Standards for School Mathematics. - If implemented at the college level, they would

establish a smooth transition between school and

college mathematics.

CRAFTY College Algebra Guidelines

- These guidelines are the recommendations of the

MAA/CUPM subcommittee, Curriculum Renewal Across

the First Two Years, concerning the nature of the

college algebra course that can serve as a

terminal course as well as a pre-requisite to

courses such as pre-calculus, statistics,

business calculus, finite mathematics, and

mathematics for elementary education majors.

Fundamental Experience

- College Algebra provides students with a college

level academic experience that emphasizes the use

of algebra and functions in problem solving and

modeling, provides a foundation in quantitative

literacy, supplies the algebra and other

mathematics needed in partner disciplines, and

helps meet quantitative needs in, and outside of,

academia.

Fundamental Experience

- Students address problems presented as real

world situations by creating and interpreting

mathematical models. Solutions to the problems

are formulated, validated, and analyzed using

mental, paper and pencil, algebraic, and

technology-based techniques as appropriate.

Course Goals

- Involve students in a meaningful and positive,

intellectually engaging, mathematical experience - Provide students with opportunities to analyze,

synthesize, and work collaboratively on

explorations and reports - Develop students logical reasoning skills needed

by informed and productive citizens

Course Goals

- Strengthen students algebraic and quantitative

abilities useful in the study of other

disciplines - Develop students mastery of those algebraic

techniques necessary for problem-solving and

mathematical modeling - Improve students ability to communicate

mathematical ideas clearly in oral and written

form

Course Goals

- Develop students competence and confidence in

their problem-solving ability - Develop students ability to use technology for

understanding and doing mathematics - Enable and encourage students to take additional

coursework in the mathematical sciences.

Problem Solving

- Solving problems presented in the context of real

world situations - Developing a personal framework of problem

solving techniques - Creating, interpreting, and revising models and

solutions of problems.

Functions Equations

- Understanding the concepts of function and rate

of change - Effectively using multiple perspectives

(symbolic, numeric, graphic, and verbal) to

explore elementary functions - Investigating linear, exponential, power,

polynomial, logarithmic, and periodic functions,

as appropriate

- Recognizing and using standard transformations

such as translations and dilations with graphs of

elementary functions - Using systems of equations to model real world

situations - Solving systems of equations using a variety of

methods - Mastering those algebraic techniques and

manipulations necessary for problem-solving and

modeling in this course.

Data Analysis

- Collecting, displaying, summarizing, and

interpreting data in various forms - Applying algebraic transformations to linearize

data for analysis - Fitting an appropriate curve to a scatterplot and

use the resulting function for prediction and

analysis - Determining the appropriateness of a model via

scientific reasoning.

- An Increased Emphasis on Pedagogy
- and
- A Broader Notion of Assessment
- Of Student Accomplishment

CRAFTY College Algebra

- Confluence of events
- Curriculum Foundations Report published
- Large scale NSF project - Bill Haver, VCU
- Availability of new modeling/application based

texts - CRAFTY responded to a perceived need to address

course and instructional models for College

Algebra.

CRAFTY College Algebra

- Task Force charged with writing guidelines
- - Initial discussions in CRAFTY meetings
- - Presentations at AMATYC Joint Math Meetings

with public discussions - - Revisions incorporating public commentary
- Guidelines adopted by CRAFTY (Fall, 2006)
- Pending adoption by CUPM (Spring, 2007)
- Copies (pdf) available at
- http//www.mathsci.appstate.edu/wmcb/ICTCM

CRAFTY College Algebra

- The Guidelines
- Course Objectives
- College algebra through applications/modeling M

eaningful appropriate use of technology - Course Goals
- Challenge, develop, and strengthen

students understanding and skills mastery

CRAFTY College Algebra

- The Guidelines
- Student Competencies
- - Problem solving
- - Functions and Equations
- - Data Analysis
- Pedagogy
- - Algebra in context
- - Technology for exploration and analysis
- Assessment
- - Extended set of student assessment tools
- - Continuous course assessment

CRAFTY College Algebra

- Challenges
- Course development
- - There are current models
- Scale
- - Huge numbers of students
- - Extraordinary variation across institutions
- Faculty development
- - Who teaches College Algebra?
- - How do we fund change?

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?

What Does the Slope Mean?

Comparison of student response on the final exams

in Traditional vs. Modeling College

Algebra/Trig Brookville College enrolled 2546

students in 2000 and 2702 students in 2002.

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 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 2000. 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.

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.

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 2000, 78 more students will

enroll at Brookville college. - 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. - 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.

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.

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 cant make their own connections with

a concept as simple as the slope of a line, they

wont 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 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

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.

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. Newtons Second Law of Motion y mx,

Einsteins formula relating energy and mass y

c2x, The ideal gas law yz nRx.

Students who see only xs and ys do not make

the connections and cannot apply the techniques

when other letters arise in other disciplines.

Should x Mark the Spot?

Keplers 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 that 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
- of Problems
- to Develop or Test for
- Conceptual Understanding

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

(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?

Two functions f and g are defined in the

following table. Use the given values in the

table to complete the table. If any entries are

not defined, write undefined.

x f(x) g(x) f(x) - g(x) f(x)/g(x) f(g(x)) g(f(x))

0 1 3

1 0 1

2 3 0

3 2 2

Two functions f and g are given in the

accompanying figure. The following five graphs

(a)-(e) are the graphs of f g, g - f, fg,

f/g, and g/f. Decide which is which.

The following table shows world-wide wind power

generating capacity, in megawatts, in various

years.

Year 1980 1985 1988 1990 1992 1995 1997 1999

Wind power 10 1020 1580 1930 2510 4820 7640 13840

(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?

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.

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

(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.

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 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) Estimate the

high temperature on March 15. (f) What are all

the dates on which the high temperature is most

likely 80??

Some 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.

Functions

- It is only in math classes that functions are

given. - Everywhere else,
- The existence of functions is observed
- Formulas for functions are created
- Functions are used to answer questions about a

context

The Need for Real-World Problems and Examples

Realistic Applications and Mathematical Modeling

- Real-world data enables the integration of data

analysis concepts with the development of

mathematical concepts and methods - Realistic applications illustrate that data arise

in a variety of contexts - Realistic applications and genuine data can

increase students interest in and motivation for

studying mathematics - Realistic applications link the mathematics to

what students see in and need to know for other

courses in other disciplines.

The Role of Technology

The Role of Technology

- Technology allows us to do many standard topics

differently and more easily. - Technology allows us to introduce new topics and

methods that we could not do previously. - Technology allows us to de-emphasize or even

remove some topics that are now less important.

Technology How?

- Students can use technology as a problem-solving

tool to - Model situations and analyze functions
- Tackle complex problems
- Students can use technology as a learning tool to
- Explore new concepts and discover new ideas
- Make connections
- Develop a firm understanding of mathematical

ideas - Develop mental images associated with abstract

concepts

Technology - Caution

- Students need to balance the use of technology

and the use of pencil and paper. - Students need to learn to use technology

appropriately and wisely.

Changing the Learning and Teaching Environment

Traditional Approach vs. Student-Centered

Approach

- With a traditional approach, students
- Listen to lectures
- Copy notes from the board
- Mimic examples
- Use technology to do calculations
- Do familiar problems in homework and on exams
- Fly through the material
- Hold instructor responsible for learning
- Go to instructor for help

Traditional Approach vs. Student-Centered

Approach

- With a student-centered approach, students
- Participate in discussions
- Work collaboratively
- Find solutions and approaches
- Use technology to investigate ideas
- Write about and use new ideas in homework and on

exams - Take time to think
- Accept responsibility for learning
- First try to help each other

Student-Centered Learning The Role of the

Instructor

- The instructor
- Designs activities
- Emphasizes learning
- Interacts with students
- Approaches ideas from the students point of view
- Controls the learning environment
- The instructor is a
- Facilitator
- Coach
- Intellectual manager

Student-Centered Learning Intended Outcomes

- Impel students to be active learners
- Make learning mathematics an enjoyable experience
- Help students develop confidence to read, write

and do mathematics - Enhance students understanding of fundamental

mathematics concepts - Increase students ability to use these concepts

in other disciplines - Inspire students to continue the study of

mathematics

- But, if college algebra and related courses

change, - what happens to the next generation of math and

science majors? - Dont they need all the traditional algebraic

skills? - But, if they dont develop conceptual

understanding and the ability to apply the

mathematics, what value are the skills?

- The Link to Calculus

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 has been steady, at best.

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.

AP Calculus

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.

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.)

Another Conclusion

It is not conscionable for departments to treat

students as mathematical cannon-fodder, by

pushing them into courses they have little hope

of surviving in order to increase the number of

sections of calculus that are offered.

Associates Degrees in Mathematics

In 2002, P There were 595,000 associate

degrees P Of these, 685 were in mathematics

This is one-tenth of one percent!

Bachelors Degrees in Mathematics

In 2002, PThere were 1,292,000 bachelors

degrees POf these, 12,395 were in mathematics

This is under one percent!

Masters Degrees in Mathematics

In 2002, PThere were 482,000 masters

degrees POf these, 3487 were in mathematics

This is 7 tenths of one percent!

PhDs Degrees in Mathematics

- In 2002,
- There were 44,000 doctoral degrees
- Of these, 958 were in mathematics
- This is just over two percent!
- But less than half were U.S. citizens

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.

The Focus in these Courses

But most college algebra courses and certainly

all precalculus courses were designed to prepare

students for calculus and most of them are still

offered in that spirit. Even though only a small

percentage of the students have any intention of

going into calculus!

A Fresh Start for Collegiate Mathematics Rethinkin

g the Courses Below Calculus MAA Notes, 2005

Nancy Baxter Hastings, et al (editors)

A Fresh Start to Collegiate Math

Introduction

Nancy Baxter Hastings Overview of the Volume

Jack Narayan Darren Narayan The Conference Rethinking the Preparation for Calculus

Lynn Steen Twenty Questions

Background

Mercedes McGowen Who are the Students Who Take Precalculus?

Steve Dunbar Enrollment Flow to and from Courses below Calculus

Deborah Hughes Hallett What Have We Learned from Calculus Reform? The Road to Conceptual Understanding

Susan Ganter Calculus and Introductory College Mathematics Current Trends and Future Directions

A Fresh Start to Collegiate Math

Refocusing Precalculus, College Algebra, and Quantitative Literacy Refocusing Precalculus, College Algebra, and Quantitative Literacy

Shelly Gordon Preparing Students for Calculus in the Twenty-First Century

Bernie Madison Preparing for Calculus and Preparing for Life

Don Small College Algebra A Course in Crisis

Scott Herriott Changes in College Algebra

Janet Andersen One Approach to Quantitative Literacy Mathematics in Public Discourse

The Transition from High School to College The Transition from High School to College

Zal Usiskin High School Overview and the Transition to College

Dan Teague Precalculus Reform A High School Perspective

Eric Robinson John Maceli The Influence of Current Efforts to Improve School Mathematics on Preparation for Calculus

A Fresh Start to Collegiate Math

The Needs of Other Disciplines The Needs of Other Disciplines

Susan Ganter and Bill Barker Fundamental Mathematics Voices of the Partner Disciplines

Rich West Skills versus Concepts

Allan Rossman Integrating Data Analysis into Precalculus Courses

Student Learning and Research Student Learning and Research

Florence Gordon Assessing What Students Learn Reform versus Traditional Precalculus and Follow-up Calculus

Rebecca Walker Student Voices and the Transition from Standards-Based Curriculum to College

A Fresh Start to Collegiate Math

Implementation

Robert Megginson Some Political and Practical Issues in Implementing Reform

Judy Ackerman Implementing Curricular Change in Precalculus A Dean's Perspective

Bonnie Gold Alternatives to the One-Size-Fits-All Precalculus/College Algebra Course

Al Cuoco Preparing for Calculus and Beyond Some Curriculum Design Issues

Lang Moore and David Smith Changing Technology Implies Changing Pedagogy

Shelly Gordon The Need to Rethink Placement in Mathematics

Influencing the Mathematics Community Influencing the Mathematics Community

Bernie Madison Launching a Precalculus Reform Movement Influencing the Mathematics Community

Naomi Fisher Bonnie Saunders Mathematics Programs for the "Rest of Us"

Shelly Gordon Where Do We Go from Here Forging a National Initiative

A Fresh Start to Collegiate Math

Ideas and Projects that Work (long papers) Ideas and Projects that Work (long papers) Ideas and Projects that Work (long papers)

Doris Schattschneider Doris Schattschneider An Alternate Approach Integrating Precalculus into Calculus

Bill Fox Bill Fox College Algebra Reform through Interdisciplinary Applications

Dan Kalman Dan Kalman Elementary Math Models College Algebra Topics and a Liberal Arts Approach

Brigette Lahme, Jerry Morris and Elias Toubassi Brigette Lahme, Jerry Morris and Elias Toubassi The Case for Labs in Precalculus

Ideas and Projects that Work (short papers) Ideas and Projects that Work (short papers) Ideas and Projects that Work (short papers)

Gary Simundza The Fifth Rule Experiential Mathematics The Fifth Rule Experiential Mathematics

Darrell Abney and James Hougland Reform Intermediate Algebra in Kentucky Community Colleges Reform Intermediate Algebra in Kentucky Community Colleges

Marsha Davis Precalculus Concepts in Context Precalculus Concepts in Context

A Fresh Start to Collegiate Math

Benny Evans Rethinking College Algebra

Sol Garfunkel From the Bottom Up

Florence Gordon Shelly Gordon Functioning in the Real World

Deborah Hughes Hallett Importance of a Story Line Functions as a Model

Nancy Baxter Hastings Using a Guided-Inquiry Approach to Enhance Student Learning in Precalculus

Allan Jacobs Maricopa Mathematics

Linda Kime Quantitative Reasoning

Mercedes McGowan Developmental Algebra The First Course for Many College Students

Allan Rossman Workshop Precalculus Functions, Data and Models

Chris Schaufele Nancy Zumoff The Earth Math Projects

Don Small Contemporary College Algebra

A Fresh Start to Collegiate Math

Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick and Sylvia Svitak Mathematics in Action Empowering Students with Introductory and Intermediate College Mathematics

Todd Swanson Precalculus A Study of Functions and Their Applications

David Wells Lynn Tilson Successes and Failures of a Precalculus Reform Project

The Need to Rethink Placement in Mathematics

Rethinking Placement Tests

- Two Types of Placement Tests
- National (standardized) tests
- Not much we can do about them.
- 2. Home-grown tests

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.

One 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.

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.

A Tale of Three Colleges in NYS

- Totally traditional curriculum developmental

through calculus. - Traditional courses developmental through

college algebra, then reform in precalculus on

up. - Totally reform developmental through upper

division offerings. - All use the same national placement test.

A Tale of Three Colleges in NYS

BUT New York State has not offered the

traditional Algebra I Geometry Algebra II

Trigonometry curriculum in over 20

years! Instead, there is an integrated

curriculum that emphasize topics such as

statistics and data analysis, probability, logic,

etc. in addition to algebra and trigonometry.

A Tale of Three Colleges in NYS

So students are being placed one, two, and even

three semesters below where they should be based

on the amount of mathematics they have

studied! And they are being punished because

of what is being assessed and what is not being

assessed, because of what was stressed in high

school and what was not stressed, because of

what was taught, not what they learned or didnt

learn.

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.

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Avg

1968 322 323 324 325 325 325 324 322 320 320 320 322 323

1969 324 324 325 326 327 326 325 323 322 321 322 324 324

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.

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?

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

Rethinking Placement Tests

What Can Be Done 1. Home-grown tests Develop

alternate versions that reflect both your

curriculum AND the different curricula that your

students have come through. 2. National

(standardized) tests Contact the test-makers

(Accuplacer ETS and Compass ACT) and lobby

them to develop alternative tests to reflect both

your curriculum and the different curricula that

your students have come through.

Why Students Take These Courses

- The vast majority of students take college

algebra and related courses because - they are required by other departments or
- they are needed to satisfy general education

requirements - As a consequence, we have to pay attention to

what the other disciplines want their students to

gain from these courses.

Connecting with Other Disciplines

All other disciplines are under pressure to teach

more material to their students, and that

material is much more than just the mathematical

ideas and applications. If we do not provide

courses that satisfy todays needs of the other

disciplines, they are likely going to drop the

requirements for our courses and include the

needed material in their own offerings.

- Voices of the Partner Disciplines
- CRAFTYs 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.

What the Physicists Said

- Courses should cover fewer topics and place

increased emphasis on increasing the confidence

and competence that students have with the most

fundamental topics.

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.

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.

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.

What Business Faculty Said

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.

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 whys not just

the hows). - Courses should stress critical thinking.
- An important student outcome is their ability to

develop appropriate models to solve defined

problems.

What Business Faculty Said

- 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 Engineers Said

- One basic function of undergraduate electrical

engineering education is to provide students with

the conceptual skills to formulate, develop,

solve, evaluate and validate physical systems.

Mathematics is indispensable in this regard.

What the Engineers Said

- The mathematics required to enable students to

achieve these skills should emphasize concepts

and problem solving skills more than emphasizing

the repetitive mechanics of solving routine

problems.

What the Engineers Said

- Students must learn the basic mechanics of

mathematics, but care must be taken that these

mechanics do not become the focus of any

mathematics course.

What the Chemists Said

- Introduce multivariable, multidimensional

problems from the outset - Listen to the equations most specific

mathematical expressions can be recovered from a

few fundamental relationships in a few steps. - Of widespread use in chemistry teaching and

research are spreadsheets to produce graphs and

perform statistical calculations

Health-Related Life Sciences

- Put special emphasis on the use of models as a

way to organize information for the purpose of

gaining insight and to provide intuition into

systems that are too complex to understand any

other way. - Students should master appropriate computer

packages, such as a spreadsheet,

symbolic/numerical computational packages

(Mathematica, Maple, Matlab), statistical

packages.

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

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

Some Implications

Although the number of college students taking

calculus is at best holding steady, the

percentage of students taking college calculus is

dropping, since overall college enrollment has

been rising rapidly. But the number of students

taking calculus in high school already exceeds

the number taking it in college. It is growing

at 8.

Some Implications

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.

Some Implications

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.

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?

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

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 men from the boys! But, can you name any

realistic applications that involve rational

functions? Why do we need them in excess?

New Visions of College Algebra

- Crauder, Evans and Noell A Modeling

Alternative to College Algebra - Herriott College Algebra through Functions and

Models - Kime and Clark Explorations in College

Algebra - Small Contemporary College Algebra

New Visions for Precalculus

- Gordon, Gordon, et al Functioning in the Real

World A Precalculus Experience, 2nd Ed - Hastings Rossman Workshop Precalculus
- Hughes-Hallett, Gleason, et al Functions

Modeling Change Preparation for Calculus - Moran, Davis, and Murphy Precalculus

Concepts in Context

New Visions for Alternative Courses

- Bennett Quantitative Reasoning
- Burger and Starbird The Heart of Mathematics

An Invitation to Effective Thinking - COMAP For All Practical Purposes
- Pierce Mathematics for Life
- Sons Mathematical Thinking

How Does the Quantitative Literacy

Initiative Relate to College Algebra?

What is Quantitative Literacy?

Quantitative literacy (QL), or numeracy, is the

knowledge and habits of mind needed to understand

and use quantitative measures and inferences

necessary to function as a responsible citizen,

productive worker, and discerning consumer. QL

describes the quantitative reasoning capabilities

required of citizens in today's information age

-- from the QL Forum White Paper

QL and the Mathematics Curriculum

The focus of the math curriculum is the

geometry-algebra-trigonometry-calculus sequence.

- In high school, the route to competitive

colleges. - The sequence is linear and hurried.
- No time to teach mathematics in contexts.
- Courses are routes to somewhere else.
- Other sequences are terminal and often second

rate.

Elements of QL

- Confidence with mathematics
- Cultural appreciation
- Interpreting data
- Logical thinking
- Making decisions

- Mathematics in context
- Number sense
- Practical skills
- Prerequisite knowledge
- Symbol sense

Two Kinds of Literacy

- Inert - Level of verbal and numerate skills

required to comprehend instructions, perform

routine procedures, and complete tasks in a

routine manner. - Liberating - Command of both the enabling skills

needed to search out information and power of

mind necessary to critique it, reflect upon it,

and apply it in making decisions.

Lawrence A. Cremins, American Education The

Metropolitan Experience 1876-1980. New York

Harper Row, 1988. (as quoted by R. Orrill in

MD)

How does the US compare to other countries?

NALS Quantitative Paradigm National Adult

Literacy Survey

Skill Level 1 - Minimal Approximate

Educational Equivalence - Dropout NALS

Competencies - Can perform a single, simple

arithmetic operation such as addition. The

numbers used are provided and the operation to be

performed is specified. NALS Examples - Total

a bank deposit entry

NALS Quantitative Paradigm

Skill Level 2 - Basic Approximate Educational

Equivalence - Average or below average HS

graduate NALS Competencies - Can perform a

single arithmetic operation using numbers that

are given in the task or easily located in the

material. The arithmetic operation is either

described or easily determined from the format of

the materials. NALS Examples - Calculate

postage and fees for certified mail - Determine

the difference in price between tickets for two

shows - Calculate the total costs of purchase

from an order form

NALS Quantitative Paradigm

Skill Level 3 - Competent Approximate Educational

Equivalence -Some postsecondary education NALS

Competencies - Can pe