voices of the partner disciplines: crafty’s curriculum foundations project

Voices of the Partner Disciplines:CRAFTY’s Curriculum

Foundations Project

College Algebra and Precalculus

In 2000, between 1,000,000 and 2,000,000 students took college algebra and precalculus courses

The focus in most of these courses is on preparing the students for calculus.

But only a small percentage ever go on to start calculus.

Enrollment Flows

Based on several studies of enrollment flows into calculus:

• Only about 10% of the students who pass 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

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

• Fitting functions to data

• Recursion and difference equations – the

mathematical language of spreadsheets

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 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.


A primary emphasis in “College Algebra” should be Mathematical Modeling:

– 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.


“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 emphasize small group projects involving inquiry and inference.


“College Algebra” courses should make appropriate use of technology to enhance conceptual understanding, visualization, and inquiry, as well as for computation

“College Algebra” courses should be student centered rather than instructor centered pedagogy: they should include hands-on activities rather than all lecture

Important Volumes

• 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.

• 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, Shelly Gordon, Flo 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.

NCTM Standards

These recommendations are clearly very much in the same spirit as the recommendations in NCTM’s Principles and Standards for School Mathematics.

If implemented at the college level, they would establish a smooth transition between school and college mathematics.

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 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 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.

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.

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 ImplicationsIf 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 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.

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 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)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?

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, f*g, f/g,

and g/f. Decide which is which.

-1

1

0 1 2 3 4 5

(a)

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5

g(x)

f(x)

-2

0

2

0 1 2 3 4 5

(b)

-2

0

2

0 1 2 3 4 5

(c)

-10

-5

0

5

10

0 1 2 3 4 5

(d)

-10

-5

0

5

10

0 1 2 3 4 5

(e)

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 13840

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?

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

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.

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) What is the most likely 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.

The Need for Real-World Problems

and Examples

Realistic Applications and Mathematical Modeling

• Genuine data enables the development of data analysis concepts to be integrated with the development of mathematical concepts

• Realistic applications illustrates 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 ofTechnology

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 student’s 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?

Don’t they need all the traditional algebraic skills?

But, if they don’t develop conceptual understanding and the ability to apply the mathematics, what value are the skills?

Calculus and Related Enrollments

In 2000, about 676,000 students took Calculus, Diff’l Eqns., Linear Algebra & Discrete Math

(This is up 6% from 1995)

In 2000, 171,400 students took one of the two AP Calculus exams – AB or BC.

(This is up 40% from 1995)

In 2002, 200,000 students took AP Calculus (Two to three times as many take the course.)

AP Calculus

Students Taking AP Calculus Exam

0

50000

100000

150000

200000

1991 1993 1995 1997 1999 2001 2003

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

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 2000,

P There were 564,933 associate degrees

P Of these, 675 were in mathematics

This is one-tenth of one percent!

Bachelor’s Degrees in Mathematics

In 2000,

PThere were 457,056 bachelor’s degrees

POf these, 3,412 were in mathematics

This is seven-tenths of one percent!

PhD’s Degrees in Mathematics

In 2000,

PThere were 44,808 doctoral degrees

POf these, 1119 were in mathematics

This is two and a half percent!

But less than half (537) were U.S. citizens

A Fresh Start for Collegiate MathematicsRethinking the Courses

Below Calculus MAA Notes, 2004

Nancy Baxter Hastings, et al(editors)

A Fresh Start to Collegiate Math

Refocusing Precalculus, College Algebra, and Quantitative Literacy

Shelly GordonPreparing 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 Harriot Changes in College Algebra

Janet AndersenOne Approach to Quantitative Literacy: Mathematics

in Public Discourse

The Transition from High School to CollegeZal 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


The Needs of Other Disciplines

Susan Ganter and Bill Barker

Fundamental Mathematics: Voices of the Partner Disciplines

Rich West Skills versus Concepts

Allan RossmanIntegrating Data Analysis into Precalculus

Courses

Student Learning and Research

Florence Gordon

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

Rebecca WalkerStudent Voices and the Transition from

Standards-Based Curriculum to College

A Fresh Start to Collegiate MathImplementati

onRobert

MegginsonSome Political and Practical Issues in Implementing

Reform

Judy AckermanImplementing Curricular Change in Precalculus: A

Dean's Perspective

Bonnie GoldAlternatives to the One-Size-Fits-All

Precalculus/College Algebra Course

Al CuocoPreparing 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

Bernie MadisonLaunching a Precalculus Reform Movement:

Influencing the Mathematics Community

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

Shelly GordonWhere Do We Go from Here: Forging a National

Initiative


Ideas and Projects that Work (long papers)Doris

Schattschneider

An Alternate Approach: Integrating Precalculus into Calculus

Bill FoxCollege Algebra Reform through

Interdisciplinary Applications

Dan KalmanElementary Math Models: College Algebra

Topics and a Liberal Arts Approach

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

Ideas and Projects that Work (short papers)

Gary Simundza The Fifth Rule: Experiential Mathematics

Darrell Abney and James Hougland

Reform Intermediate Algebra in Kentucky Community Colleges

Marsha Davis Precalculus: Concepts in Context


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


Ernie Danforth, Brian Gray, Arlene Kleinstein, Rick Patrick and

Sylvia Svitak

Mathematics in Action: Empowering Students with Introductory and Intermediate College Mathematics

Todd SwansonPrecalculus: A Study of Functions and Their Applications

David WellsLynn Tilson

Successes and Failures of a Precalculus Reform Project

The Need to RethinkPlacement

in Mathematics

Rethinking Placement Tests

Two Types of Placement Tests:

1. National (standardized) testsNot much we can do about them.

2. Home-grown tests

Rethinking Placement TestsFour scenarios:

1. Students come from traditional curriculum into traditional curriculum.

2. Students from Standards-based curriculum into traditional curriculum.

3. 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

1. Totally traditional curriculum – developmental through calculus.

2. Traditional courses – developmental through college algebra, then reform in precalculus on up.

3. Totally reform – developmental through upper division offerings.

All use the same national placement test.


BUTNew 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.


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

didn’t 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:

0.03923( ) 1.656 299.5tF t e

2. They then create a sinusoidal model to fit the monthly oscillatory behavior about the exponential curve

13.5sin 2 0.5

24S t t

3. They then combine the two components to get

0.03923 11.656 3.5sin 2 299

24tC t F t S t e t

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) testsContact 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.

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 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.


• 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.


• 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.


• 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.


• 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.

CHEMISTRY

• 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…graph & stat

CHEMICAL ENGINEERING

• “A good chemical engineer brings together the fundamentals to build and refine a mathematical model of a process that will help him or her understand and optimize its performance.”

• “Alumni surveys typically show that probability and statistics, in addition to the extensive use of spread-sheeting software, is the most common application of mathematics for the practicing chemical engineer with a B. S. degree.”

CIVIL ENGINEERING• Modeling the 2nd component in a 5-component problem solving

scheme• “…adjust the balance of course material to more applied and

numerical solution techniques and less advanced analytical techniques, integrating more technology into the curriculum, and coordinating the mathematics and engineering curricula.”

• Introductory mathematics courses to be taken over a 3 or 4 year period rather than 2 years

• “Technology must be a major component of mathematics curriculum”. Use Microsoft Excel and MathCad, “technology used in their engineering classes and in their careers.”

ELECTRICAL ENGINEERING

• “Be able to mathematically model physical reality”-Outcome 6 of 8 for math to EE

• “Of critical interest are the logical thinking skills …” “ mathematics required should emphasize concepts and problem-solving skills more than emphasizing repetitive mechanics of solving routine problems.

• “Introducing symbolic manipulation programs, e.g. MathCAD, Mathematica, Maple, would be valuable to subsequent EE courses”-pedagogy choices need to be coordinated.

MECHANICAL ENGINEER

Our “we want” list of 9 math abilities includes:

• “Use modeling techniques”.

• “Be familiar with software applications for numerical and symbolic computation. They also need to be introduced to spreadsheets.”

• “The internet gives access to vast amounts of information, allowing solution of more interesting complex problems.”

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.

PHYSICS• “The panel was unanimous in suggesting that practice

in solving problems should be extensively couched in real-world contexts that are meaningful to students.”

• “There is general agreement among the panelists that some experience with a symbolic manipulation program like Mathematica or Maple is desirable.

• “…a compelling alternative opinion: Spread sheets are by far the best medium (for teaching with technology) since data, text, and graphics are all visible at once, and since the techniques are easily learned and useful for numerical approximations “really try to understand what is in the mind of the student”-best practice in instruction for all partners.

Teacher Prep: K-12 Mathematics• “In general, there should be less symbolic manipulation,

and more modeling and problem solving from both a discrete and continuous perspective” in algebra

• “There was virtually universal agreement among workshop participants that reasoning and proof should be a theme in all college mathematics courses, beginning at least with calculus.”

• In introductory courses, ask students to explain their thinking or to justify their responses based on definitions.

• “Spread sheets and the graphing and table-generating features of graphing calculators can be effectively used to solve problems about functions and families of functions.”

Some Major Themes in Technology

1. Math Courses as FiltersWe regret that, in many settings, mathematics courses are intentionally viewed as filters for entry into IT curricula.

Information TechnologyMathematics should not be a filter that causes students to drop out of our programs.

Biotechnology and Environmental TechnologyThese students get discouraged in remedial mathematics, and a refresher class might be more effective.

Mechanical and Manufacturing Technology


2. Who should teach technical math courses?Should mathematics be taught in the technical program itself, or should it be taught by the mathematics department? Should it be taught by “captive” mathematics instructors who work for the technical program?

Biotechnology and Environmental TechnologyThere needs to be more cooperation … between faculty members in mathematics and … the technical fields. … Arrange for interdepartmental visits. … [Invite industry representatives to make presentations.]

Electronics and Telecommunications


2. Who should teach technical math courses (ctd)?Team teaching, with mathematics faculty entering apprenticeships with technology faculty, would improve the teaching of mathematics … required in the content area.


One might ask whether some mathematical reasoning skills could be obtained from other disciplines. … we promote the notion of mathematics across the curriculum.

Information Technology


3. Education of Mathematics FacultyMathematics instructors should be encouraged to attend the technical courses.


Participation in summer internships in industry is an excellent way for teachers to understand the practical aspects of mathematics in the IT field.


Mathematics instructors may need to apprentice in the technology areas to learn the applications and understand which topics are important.



4. Course ContentMastery of the basics is more important than exposure to a lot of mathematics.

Biotechnology and Environmental TechnologySemiconductor technicians need little more than ratios, elementary algebra, and elementary statistics. Electronics and telecommunications technicians need more mathematics.

Electronics and TelecommunicationsPreparation for students pursuing IT careers should not require advanced topics but should instead provide a solid foundation of elementary content.



5. Applications-based coursesCourses in algebra and statistics should use an applications-based

approach instead of the traditional textbook approach, considering real-life problems.Biotechnology and Environmental Technology

IT-based applications should drive the development of mathematical theory and its use. … Applications should be considered first, and then theory.Information Technology

Avoid teaching mathematics to simply teach mathematics. Move heavily to the application of the concepts within the mathematics classroom.Mechanical and Manufacturing Technology


6. Technology in the Mathematics ClassroomStudents should develop proficiency with at least one [computation software] program, as well as a working knowledge of spreadsheets.


Students should use a variety of software packages in mathematics classes. … students should be able to use ordinary and statistical calculators and should become comfortable with spreadsheets for calculation and graphing.

Biotechnology and Environmental Technology


6. Technology in the Math Classroom (ctd)To the extent that mathematics courses teach and reinforce the use of technological tools, IT students are well served.


Our students should have experience with graphing calculators, word processors, spreadsheets, data base management software, and computer presentation applications … they should also have experience with mathematical and statistical software.



7. Estimation and Approximation[Students] should question their answers, not accept them blindly. Students should use common sense and estimation skills.

Biotechnology and Environmental TechnologyStudents should learn how to perform an error analysis. They must learn to estimate answers and to obtain approximate solutions.

Electronics and Telecommunications[Technological] tools should never replace abilities such as estimating, performing simple mental arithmetic with precision, or evaluating a tool’s accuracy.

Information TechnologyApproximation and estimation: having a feel for the right order of magnitude, the right units of measure, and the appropriate precision for an answer.



8. Articulation with Four-Year SchoolsIf mathematics courses were developed for specific applications, they would probably not be transferable towards a baccalaureate degree program.

Biotechnology and Environmental TechnologyWe focus on the mathematical skills students should master while completing the associate degree as an entry into the job market. … It may be necessary for students to complete a bridge course in order to enter a baccalaureate program.

Information TechnologyProvide bridge courses for job-oriented students in two-year programs to help them make the jump to a baccalaureate program.



9. The Most Frequently Requested Topics

Basic algebra

Basic statistics

Graphs and graphical representation

Spreadsheets

Software packages


10. The Most Frequently Requested Skills

Problem-solving

Teamwork

Communication

Estimation

Multi-step projects

CUPM Curriculum Guide 2004

• “Indeed, the participants in the Curriculum Foundations workshops were so excited by the possibility of increasing the use of real models in mathematics courses that many volunteered to help develop such models”

• “…chemistry gave an urgent plea for an earlier introduction to multivariable calculus…Even more colleagues expressed dissatisfaction with the level of student understanding of concepts-geometric and otherwise-in three dimensions.”

CUPM CG ‘04 Thinking Theme

• “Faculty who participated in the CF workshops commented frequently that all disciplines look to mathematics courses to enhance students’ abilities to reason logically and deductively, but that they want this ability developed in a context that increases understanding of underlying concepts.”

• “…the computer scientists specifically requested that lower-division discrete mathematics courses include an introduction to formal proof”

• “The correct balance can and should be determined through consultation with colleagues in partner disciplines.”

CUPM CG ’04 Technology Use• “Participants in the CF workshops understood the

importance of technology, and took for granted that mathematics courses would incorporate technology to some degree”…. “Perhaps more surprising is that spreadsheets are the most utilized technology for a large number of partner disciplines”(p. 36).

• Recommendation 5 of 6 says every level of the curriculum should have some courses incorporating technology use. Cross reference with 2001 Guidelines for Programs and Departments (p 7).

RESOURCES

• Today’s slides at

www.piercecollege.edu/faculty/yhoshibw

• CG at maa.org/cupm/curr_guide.html

• CF at maa.org/cupm/crafty/cf_project.html

• Instructor Resources at maa.org/cupm/illres_refs.html

Descriptions and web links to examples, experiences, and resources for one to work toward the recommendations of CG

• MET at cbmsweb.org

• MAA Guide-Prog./ Dept. maa.org/guidelines

Some Implications

Although the number of students taking calculus in college is holding steady, the situation is actually worse – the percentage of students taking calculus is dropping, since overall college enrollment has been rising rapidly.

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.

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:

22 1

cos1 tan

xx

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 fractions–often 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

InitiativeRelate 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

• 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.

The focus of the math curriculum is the geometry-algebra-trigonometry-calculus

sequence.

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 M&D)

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 - BasicApproximate 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 ParadigmSkill Level 3 - CompetentApproximate Educational Equivalence -Some postsecondary educationNALS Competencies -Can perform tasks where two or more numbers are needed to solve the problem and they must be found in the material. The operation(s) needed can be determined from the arithmetic relation terms used in the question or directive.NALS Examples - Use a calculator to calculate the difference between the regular and sale price- Calculate miles per gallon from information on a mileage record chart - Use a calculator to determine the discount from an oil bill if paid within 10 days

NALS Quantitative ParadigmSkill Level 4 - AdvancedApproximate Educational Equivalence -Bachelor’s or advanced degreeNALS Competencies -Can perform two or more operations in sequence or a single operation in which the quantities are found in different types of displays, or where the operations must be inferred from the information given or from prior knowledge.NALS Examples- Determine the correct change using information in a menu-Calculate how much a couple would receive from Supplemental Security Income, using an eligibility pamphlet- Use information stated in a news article to calculate the amount of money that should go to raising a child

NALS Quantitative ParadigmSkill Level 5 - SuperiorApproximate Educational Equivalence -High achieving college-educated populationsNALS Competencies -Can perform multiple operations sequentially, and can also find the features of problems embedded in text or rely on background knowledge to determine the quantities or operations needed.NALS Examples- Use a calculator to determine the total cost of carpet to cover a room- Use information in a news article to calculate the difference in time for completing a race- Determine shipping and total costs on an order form for items in a catalog

Source: USDOE, NCES, National Adult Literacy Survey, 1992, in Literacy in the Labor Force: Results from the NALS, September 1999, p. 61.

Many College Graduates Demonstrate Weak Quantitative Literacy Skills

Grads:2 Yr. Colleges

Grads:4 Yr. Colleges

Level 5: High 5 13

Level 4 30 40

Level 3 44 40

Level 2 17 10

Level 1: Low 4 3

Quantitative Literacy and Job Opportunity, 1998-2008

Quantitative

Skill Level

Mathematical Literacy Level

15% of the Labor Force

9% of New Jobs, 1998-2008 1998 Earnings: $20,300

12% of All Jobs in 2008

(Below Average H. S. Graduate) 24% of the Labor Force






Minimal(Dropout)

Basic

Competent

(Some Postsecondary)

Advanced/Superior(Bachelor’s Degree)


33% of New Jobs, 1998-2008 1998 Earnings: $45,40027% of All Jobs in 2008

Source: National Adult Literacy Survey; Current Population Survey; Bureau of Labor Statistics Employment Projections, 1998-2008.

Where Does College Algebra Fit In?

QL is something that should permeate the entire mathematics curriculum, so that every student develops these skills.

The one existing course that provides the best opportunity to stress QL is college algebra:• It has the largest enrollment• It does not prepare or motivate large numbers of students to go on to calculus• It is taken to prepare students for courses in other disciplines, and the themes of QL are the mathematical topics needed in most other disciplines today.

SomeSample

Programs

A Sample Program

•

The Challenges to Be Faced

The Challenges Ahead

• Convincing the math community

1. Conducting a large tracking study 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.


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

• Convincing academic bodies outside of mathematics to allow alternatives to traditional college algebra courses to fulfill general education requirements.


• Convincing statewide agencies and legislative bodies to change general education requirements to allow alternatives to traditional college algebra courses.


• 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.


• Gaining the active support of representatives of a wide variety of other disciplines in the effort to refocus the courses below calculus.

• Gaining the active support of representatives of business, industry, and government in this initiative.


• Developing a faculty development program to assist faculty, especially part time faculty and graduate TA’s, to teach the new versions of these courses.

• Influencing teacher preparation programs to rethink the courses they offer to prepare the next generation of teachers in the spirit of this initiative.


• 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 implementation of existing “reform” versions of these courses.

voices of the partner disciplines: crafty’s curriculum foundations project

Documents