Deborah Hughes Hallett
University of Arizona
Harvard University

Why Change? A US-European Perspective
•
Role of Mathematics and Statistics is Changing:
– More fields require more mathematics (eg bioinformatics, finance)
– Business and government policy require data analysis for sound decision-making
•
Technology and the Internet Changes the Way
Mathematics and Statistics are Done:
– Mathematica, Excel, statistical software, etc
–
Business and industry run on technology
– Data is much more readily available
•
Students are Changing:
– Expect to see how mathematics is related to their field of interest.
Expect to use technology
– Don’t learn well in passive lectures

To Enable Students to Use Their
Mathematics in Other Settings
•
Mathematics needs to be taught showing its connections to other fields
–
Otherwise students think of it as unrelated
•
Problems are needed that probe student conceptual understanding
– Otherwise some students only memorize

Changes Currently Underway
• Curriculum:
– Multiple representations: “Rule of Four”
–
More explicit intellectual connections to other fields
•
Pedagogy:
–
More active: Group work, projects
– More emphasis on interpretation and understanding
• Technology:
– Reflects professional practice (where possible)
–
Enables more realistic problems
Changes affect calculus, differential equations, statistics, linear algebra, and quantitative reasoning

Most Significant Change Made:
Types of Problems Given
Problems are important because they tell us what our students know
• Problems should test understanding as well as computational skill
• What do these problems look like?
Examples follow from Calculus, 4 th edn, by
Hughes-Hallett, Gleason, McCallum, et al.
•
Many use “Rule of Four”

Translating between representations promotes understanding
• Symbolic:
Ex: What does the form of a function represent?
•
Graphical:
Ex: What do the features of the graph convey?
•
Numerical:
Ex: What trends can be seen in the numbers?
•
Verbal:
Ex: Meaning is usually carried by words or pictures

New problem types: Interpretation of the derivative from Calculus, 4th edn, by Hughes-Hallett, Gleason, McCallum, et al.

Interpretation: Graphs
The graphs show the temperature of potato put in an oven at time x = 0. Which potato
(a) Is in the warmest oven?
(b) Started at the lowest temperature?
(c) Heated up fastest?

•
Previously, until early 1990s:
50+ exercises to graph functions like
•
•
Occasional “proofs”: really calculations with answer given
No applications

from Calculus, 4th edn, by Hughes-Hallett, Gleason, McCallum, et al.

•
Previously, until early 1990s:
60+ exercises deciding whether a series with a given formula converges. Only variable is x . Could be done without understanding what convergence means
• No graphical, numerical problems.
• Few applications.

New problem: Linear Approximation
The figure shows the tangent line approximation to f ( x ) near x = a .
•
Find a , f ( a ), f’
( a ).
•
Estimate f (2.1) and f (1.98). Are these under- or overestimates? Which would you expect to be most accurate?
from Calculus, 4th edn, by Hughes-Hallett, Gleason, McCallum, et al.

Newer: Application of Taylor series
(Calculus 4th edn, p.516 Problem 36.)

Project: Differential Equations from Calculus, 4 th edn, by Hughes-Hallett, Gleason, McCallum, et al.
PREVENTING THE SPREAD OF AN
INFECTIOUS DISEASE
There is an outbreak of the disease in a nearby city. As the mayor, you must decide the most effective policy for protecting your city:
I.
Close off the city from contact with the infected region. Shut down roads, airports, trains, busses, and other forms of direct contact.
II.
Install a quarantine policy. Isolate anyone who has been in contact with an infected person or who shows symptoms of the disease.

SARS in Hong Kong: No quarantine
Analyzed using 2003 World Health Organization data from Hong Kong

SARS in Hong Kong: With quarantine
Analyzed Using 2003 World Health Organization data from Hong Kong

How Widespread are these Changes?
Example: Calculus in US
Universities:
– Most universities have experimented with new syllabi, technology; some have changed their courses significantly
End of High School Exam (AP Exam) taken by
200,000 students a year:
– New syllabus with more focus on big ideas; less on list of problem types.
Uses graphing calculators,
–
National Academy of Science study “Learning for Understanding” supported new syllabus.
International IB Exam:
– Made similar changes

How Successful are These Changes?
Example of Evaluation: Results with ConcepTests
(Conceptual questions; Active Learning)
Conceptual questions
Standard computational problems
With ConcepTests 73% 63%
Standard Lecture 17% 54%

Increasing Diversity of Student
Backgrounds and Interests
Increasing Demands from Other
Fields, Business, and Industry
Computer Algebra Systems (CAS)
And?? What are Your Ideas??

• Many of the changes in the teaching of mathematics over last decade were initiated by people actively involved in the classroom.
• This is why we are here; I am looking forward to learning from all of you in this conference