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					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)
2003 AP                                       Graph of f '
Calculus exam
AB4/BC4



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                                     Graph of f '
Calculus exam
AB4/BC4



Let f be a function defined on the closed interval  3  x  4 with
 What about: “At x = 2 because it is the location of a local
 f 0   3. The graph of f ', the derivative of f , consists of one line
 maximum of the graph of f .”? Does this necessarily imply
 a point of a semicircle, the graph
segment andinflection of as shown. 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.
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
1   AP, IB, dual enrollment, transfer credit)
    ~300,000 retake calculus taken in HS

2      Some start by retaking the calculus they
       studied in high school

3      Some are required to take precalculus first
4   ~200,000 will take calculus for first time
 4 ~200,000 will take calculus for first time


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
3
       Some are required to take precalculus first


  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
4 challenging and engaging them.
    ~300,000 retake calculus taken in HS
2
        Some start by retaking the calculus they
        studied in high school


  We need a better solution than having these students
3 retread familiar territory, but at a much faster pace, in
  larger classes, and with an instructor who is unable
4
  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
1   AP, IB, dual enrollment, transfer credit)
    ~300,000 retake calculus taken in HS
  These are our success stories but:
2        Some start by retaking the calculus they
  •We need to worry about articulation with their
         studied in high school
  high school experience.
3        Some are required to take precalculus first
  •We need to work at both challenging and
    ~200,000 will take calculus for first time
4 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?
(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   x 2 sin 1 / x   2 , x  0, g 0   0.
       x  0  g x   0.
                       h 2 sin 1 / h   2  0
      g ' 0   lim                                 0.
                h0            h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   x 2 sin 1 / x   2 , x  0, g 0   0.
       x  0  g x   0.
                       h 2 sin 1 / h   2  0
      g ' 0   lim                                 0.
                h0            h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   x 2 sin 1 / x   2 , x  0, g 0   0.
       x  0  g x   0.
                       h 2 sin 1 / h   2  0
      g ' 0   lim                                 0.
                h0            h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.
   This PowerPoint will be available at
   K-12 education, it is the goal.
In www.macalester.edu/~bressoud/talks
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.

				
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