The Teaching of Mathematics: The Teaching of Mathematics: What Changes are on the What Changes are on the

Horizon?Horizon?

Deborah Hughes HallettDeborah Hughes Hallett

University of ArizonaUniversity of Arizona

Harvard UniversityHarvard University

Why Change? A US-European PerspectiveWhy Change? A US-European Perspective

• Role of Mathematics and Statistics is Changing: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 Technology and the Internet Changes the Way Mathematics and Statistics are Done: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: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 To Enable Students to Use Their Mathematics in Other SettingsMathematics in Other Settings

• Mathematics needs to be taught showing its Mathematics needs to be taught showing its connections to other fieldsconnections to other fields– Otherwise students think of it as unrelatedOtherwise students think of it as unrelated

• Problems are needed that probe student Problems are needed that probe student conceptual understanding conceptual understanding – Otherwise some students only memorizeOtherwise some students only memorize

Changes Currently UnderwayChanges Currently Underway

• Curriculum: Curriculum: – Multiple representations: “Rule of Four”– More explicit intellectual connections to other fields

• Pedagogy:Pedagogy:– More active: Group work, projects– More emphasis on interpretation and understanding

• Technology:Technology:– Reflects professional practice (where possible)– Enables more realistic problems

Changes affect calculus, differential equations, Changes affect calculus, differential equations, statistics, linear algebra, and quantitative reasoningstatistics, 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, 4th edn, by Hughes-Hallett, Gleason, McCallum, et al.

•Many use “Rule of Four”

Rule of Four: Translating between representations Translating between representations

promotes understandingpromotes understanding

• SymbolicSymbolic: Ex: What does the form of a function represent?

• GraphicalGraphical: Ex: What do the features of the graph convey?

• NumericalNumerical: Ex: What trends can be seen in the numbers?

• VerbalVerbal: Ex: Meaning is usually carried by words or pictures

New problem types: Interpretation of New problem types: Interpretation of the derivativethe derivative

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

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?

Interpretation: GraphsInterpretation: Graphs

Previously, until early 1990s:• 50+ exercises to graph functions like

• Occasional “proofs”: really calculations with answer given

• No applications

How Has Graphing Changed?How Has Graphing Changed?

Newer: Graphing with ParametersNewer: Graphing with Parameters

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

How Have Infinite Series Changed?How Have Infinite Series Changed?

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 ApproximationNew problem: Linear Approximation

• 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?

The figure shows the tangent line approximation to f(x) near x = a.

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

Newer: Application of Taylor seriesNewer: Application of Taylor series

(Calculus 4th edn, p.516 Problem 36.)

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.

Project: Differential Equations from Calculus, 4th edn, by Hughes-Hallett, Gleason, McCallum, et al.

SARS in Hong Kong: No quarantine

Analyzed using 2003 World Health Organization data from Hong KongAnalyzed using 2003 World Health Organization data from Hong Kong

SARS in Hong Kong: With quarantine

Analyzed Using 2003 World Health Organization data from Hong KongAnalyzed Using 2003 World Health Organization data from Hong Kong

How Widespread are these Changes?How Widespread are these Changes?Example: Calculus in USExample: Calculus in US

Universities:Universities: –Most universities have experimented with new syllabi, technology; some Most universities have experimented with new syllabi, technology; some have changed their courses significantly have changed their courses significantly

End of High School Exam (AP Exam) taken by End of High School Exam (AP Exam) taken by 200,000 students a year:200,000 students a year:–New syllabus with more focus on big ideas; less on list of problem types. New syllabus with more focus on big ideas; less on list of problem types. Uses graphing calculators,Uses graphing calculators,–National Academy of Science study “Learning for Understanding” National Academy of Science study “Learning for Understanding” supported new syllabus. supported new syllabus.

International IB Exam:International IB Exam: –Made similar changesMade similar changes

Example of EvaluationExample of Evaluation: Results with ConcepTests : Results with ConcepTests (Conceptual questions; Active Learning)(Conceptual questions; Active Learning)

Conceptualquestions

Standard computational

problems

With ConcepTests 73% 63%

Standard Lecture 17% 54%

How Successful are These Changes?How Successful are These Changes?

Challenges of FutureChallenges of Future

Increasing Diversity of Student Backgrounds and Interests

Increasing Demands from Other Fields, Business, and Industry

Computer Algebra Systems (CAS)

And?? What are Your Ideas??

How Such Challenges are Met:How Such Challenges are Met:

• Many of the changes in the teaching of Many of the changes in the teaching of mathematics over last decade were mathematics over last decade were initiated by people actively involved in initiated by people actively involved in the classroom. the classroom.

• This is why we are here; I am looking This is why we are here; I am looking forward to learning from all of you in forward to learning from all of you in this conferencethis conference

Thank You!Thank You!