The Teaching of Mathematics: What Changes are on the Horizon? Deborah Hughes Hallett University of Arizona Harvard University

Why 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 Mathematics and Statistics are Done: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: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 fieldsMathematics needs to be taught showing its connections to other fields –Otherwise students think of it as unrelated Problems are needed that probe student conceptual understandingProblems are needed that probe student conceptual understanding –Otherwise some students only memorize

Changes 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, 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 Rule of Four: Translating between representations promotes 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 the 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: Graphs

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

Newer: Graphing with Parameters from Calculus, 4th edn, by Hughes-Hallett, Gleason, McCallum, et al.

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

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

Example of Evaluation: Results with ConcepTests (Conceptual questions; Active Learning) Conceptual questions Standard computational problems With ConcepTests73%63% Standard Lecture17%54% How Successful are These Changes?

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