David Bressoud, Macalester College

Visualizing the Future of Mathematics Education, USC, April 13, 2007

This PowerPoint will be available at www.macalester.edu/~bressoud/talks

80% precalculus and precollege

53% introductory and precollege

53% introductory and precollege,

72% if you count Calculus I as high school math

Over the past quarter century, total enrollment has increased 42%.

College faculty cannot afford to ignore what is happening in K-12 education.

Initiatives to clarify expectations:

•NCTM Focal Points

•College Board Standards for College Success: Mathematics and Statistics

•Achieve, Inc. (National Governor’s Association), Secondary Mathematics Expectations

1. Calculus Reform Reaches Maturity

2. Challenges of the Movement of Calculus into High School

3. Conclusion

January, 1986, Tulane — What has happened since?

January, 1986, Tulane — What has happened since?

•Major NSF calculus initiative

January, 1986, Tulane — What has happened since?

•Major NSF calculus initiative

•Noticeable change in the texts:

Symbolic, graphical, numerical, written representations

Incorporation of calculator and computer technology

More varied problems, opportunities for exploration in depth

January, 1986, Tulane — What has happened since?

•Major NSF calculus initiative

•Noticeable change in the texts:

Symbolic, graphical, numerical, written representations

Incorporation of calculator and computer technology

More varied problems, opportunities for exploration in depth

•No noticeable shift in the syllabus

January, 1986, Tulane — What has happened since?

•Major NSF calculus initiative

•Noticeable change in the texts:

Symbolic, graphical, numerical, written representations

Incorporation of calculator and computer technology

More varied problems, opportunities for exploration in depth

•No noticeable shift in the syllabus

•Advanced Placement Calculus embraced calculus reform in mid-1990s (Kenelly, Kennedy, Solow, Tucker)

Graph of f '

Let f be a function defined on the closed interval −3≤x≤4 with

f 0( ) =3. The graph of f ', the derivative of f , consists of one line segment and a semicircle, as shown.

(b) Find the x-coordinate of each point of inflection of the graph of f on the open interval −3 < x< 4. Justify your answer.

2003 AP Calculus exam

AB4/BC4

Graph of f '

Let f be a function defined on the closed interval −3≤x≤4 with

f 0( ) =3. The graph of f ', the derivative of f , consists of one line segment and a semicircle, as shown.

(b) Find the x-coordinate of each point of inflection of the graph of f on the open interval −3 < x< 4. Justify your answer.

2003 AP Calculus exam

AB4/BC4

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

At 5-year intervals starting in 1990, CBMS has been tracking number of sections of mainstream Calculus I that use various markers of reform calculus:

•Use of graphing calculators

•Use of computer assignments

•Use of writing assignments

•Use of group projects

Results from 2005 are just in.

Use of online resources in mainstream Calculus I (2005):

PhD: 9%

MA: 2%

BA: 2%

2-year: 5%

AP Calculus currently growing at >14,000/year (about 6%)

AP Calculus currently growing at >14,000/year (about 6%)

Estimated # of students taking Calculus in high school (NAEP, 2005): ~ 500,000

Estimated # of students taking Calculus I in college: ~ 500,000

(includes Business Calc)

~200,000 arrive with credit for calculus (includes AP, IB, dual enrollment, transfer credit)

~300,000 retake calculus taken in HS

Some start by retaking the calculus they studied in high school

Some are required to take precalculus first

~200,000 will take calculus for first time

~200,000 arrive with credit for calculus (includes AP, IB, dual enrollment, transfer credit)

~300,000 retake calculus taken in HS

Some start by retaking the calculus they studied in high school

Some are required to take precalculus first

~200,000 will take calculus for first time

1

2

3

4

~200,000 will take calculus for first time

4

Increasingly, these are students who neither need nor want more than a basic introduction to calculus (i.e. Biology majors). Challenge is to give them a one-semester course that

•Acknowledges that they may not be our strongest students, but

•Builds their mathematical skills,

•Gives them an understanding of calculus, and

•Does not cut them off from continuing the study of calculus

~300,000 retake calculus taken in HS

Some are required to take precalculus first

3

4

We need a better solution for these students. Again, the challenge is to give them a course that enables them to overcome their deficiencies while challenging and engaging them.

~300,000 retake calculus taken in HS

Some start by retaking the calculus they studied in high school

2

3

4

We need a better solution than having these students retread familiar territory, but at a much faster pace, in larger classes, and with an instructor who is unable to give them the individual attention that they experienced when they struggled with these ideas the previous year.

One approach (Macalester):

Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors:

•Understanding of rates of change, meaning of derivative

One approach (Macalester):

Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors:

•Understanding of rates of change, meaning of derivative

•Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers

One approach (Macalester):

Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors:

•Understanding of rates of change, meaning of derivative

•Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers

•Reading, writing, and finding numerical solutions to differential equations and systems of diff eqns

One approach (Macalester):

Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors:

•Understanding of rates of change, meaning of derivative

•Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers

•Reading, writing, and finding numerical solutions to differential equations and systems of diff eqns

•Integration as limit of sum of product and as anti-derivative

One approach (Macalester):

Replace traditional Calculus I with Applied Calculus. Syllabus is designed to stand on its own and is built on what students really need for non-mathematical majors:

•Understanding of rates of change, meaning of derivative

•Functions of several variables, partial and directional derivatives, geometric meaning of Lagrange multipliers

•Reading, writing, and finding numerical solutions to differential equations and systems of diff eqns

•Integration as limit of sum of product and as anti-derivative

•Enough linear algebra to understand geometric interpretation of linear regression.

~200,000 arrive with credit for calculus (includes AP, IB, dual enrollment, transfer credit)

~300,000 retake calculus taken in HS

Some start by retaking the calculus they studied in high school

Some are required to take precalculus first

~200,000 will take calculus for first time

1

2

3

4

These are our success stories but:

•We need to worry about articulation with their high school experience.

•We need to work at both challenging and enticing these students.

Dual enrollment

In spring, fall 2005 (combined), 33,436 students studied Calculus I under dual enrollment programs: 14,030 in connection with 4-year colleges, 19,406 in connection with 2-year colleges.

Control of 4-year colleges 2-year colleges

syllabus 92% 89%textbook 44% 74%instructor 48% 52%final exam 30% 37%

BC exam,

8818 in 2002

13,809 in 2006

57% increase

Need for curricula that engage and entice

E.g., Approximately Calculus, Shariar Shariari, AMS, 2006

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

What do we mean by “point of inflection”?

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

What do we mean by “point of inflection”?

What do we mean by “concavity”?

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

What do we mean by “point of inflection”?

What do we mean by “concavity”?

Graph of f is concave up on [a,b] if every secant line lies above (> or ≥ ?) the graph of f.

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

What do we mean by “point of inflection”?

What do we mean by “concavity”?

Graph of f is concave up on [a,b] if every secant line lies above (> or ≥ ?) the graph of f. Graph of f ' is increasing over some interval with right-hand endpoint at 2, decreasing over interval with left-hand endpoint at 2.

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

Is it possible to have g(x) < g(2) for all x < 2, but on every interval with right-hand endpoint at 2, there is a subinterval over which g is strictly decreasing?

g x( ) =x2 sin 1 / x( )−2( ), x≠0, g 0( ) =0.

x≠0 ⇒ g x( ) < 0.

g' 0( ) =limh→ 0

h2 sin 1 / h( )−2( )−0h−0

=0.

g' x( ) =2x sin 1 / x( )−2( )−cos 1 / x( ),

−6 x −cos 1 / x( ) ≤g' x( ) ≤6 x −cos 1 / x( ).

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

g x( ) =x2 sin 1 / x( )−2( ), x≠0, g 0( ) =0.

x≠0 ⇒ g x( ) < 0.

g' 0( ) =limh→ 0

h2 sin 1 / h( )−2( )−0h−0

=0.

g' x( ) =2x sin 1 / x( )−2( )−cos 1 / x( ),

−6 x −cos 1 / x( ) ≤g' x( ) ≤6 x −cos 1 / x( ).

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

g x( ) =x2 sin 1 / x( )−2( ), x≠0, g 0( ) =0.

x≠0 ⇒ g x( ) < 0.

g' 0( ) =limh→ 0

h2 sin 1 / h( )−2( )−0h−0

=0.

g' x( ) =2x sin 1 / x( )−2( )−cos 1 / x( ),

−6 x −cos 1 / x( ) ≤g' x( ) ≤6 x −cos 1 / x( ).

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

No.

(b) Find the x-coordinate of each point of inflection of the graph of

f on the open interval −3 < x< 4. Justify your answer.

What about: “At x = 2 because it is the location of a local maximum of the graph of f .”? Does this necessarily imply a point of inflection of the graph of f ?

No.

What actually happened: Stem describes the graph of the derivative of f as consisting of a line segment and a semi-circle. General policy is to allow students to assume any information given in the stem without needing to reference it explicitly. In this case, the answer is Yes. Student received credit for the second point.

Whether or not it should be, calculus is the linchpin of the entire mathematics curriculum.

In K-12 education, it is the goal.

In undergraduate and graduate mathematics, it is the foundation.

Whether or not it should be, calculus is the linchpin of the entire mathematics curriculum.

In K-12 education, it is the goal.

In undergraduate and graduate mathematics, it is the foundation.

The movement of calculus into the high school curriculum is causing strains that must be better understood and that can only be addressed by a serious and thorough reappraisal of what we teach as well as how we teach it.

Whether or not it should be, calculus is the linchpin of the entire mathematics curriculum.

In K-12 education, it is the goal.

In undergraduate and graduate mathematics, it is the foundation.

The movement of calculus into the high school curriculum is causing strains that must be better understood and that can only be addressed by a serious and thorough reappraisal of what we teach as well as how we teach it.

Such a reappraisal will have repercussions throughout the curriculum, spreading in both directions.

Whether or not it should be, calculus is the linchpin of the entire mathematics curriculum.

In K-12 education, it is the goal.

In undergraduate and graduate mathematics, it is the foundation.

The movement of calculus into the high school curriculum is causing strains that must be better understood and that can only be addressed by a serious and thorough reappraisal of what we teach as well as how we teach it.

Such a reappraisal will have repercussions throughout the curriculum, spreading in both directions.

This PowerPoint will be available at www.macalester.edu/~bressoud/talks