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

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

10. 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

11. 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

12. 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

13. 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”

14. 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

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

16. 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?

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

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

19. 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.

20. 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.

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

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

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

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

25. 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

26. How Successful are These Changes? Example of Evaluation: Results with ConcepTests (Conceptual questions; Active Learning)

27. 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??

28. 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. • This is why we are here; I am looking forward to learning from all of you in this conference Thank You!