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# The Teaching of Mathematics: What Changes are on the Horizon - PowerPoint PPT Presentation

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: More fields require more mathematics (eg bioinformatics, finance)

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

Deborah Hughes Hallett

University of Arizona

Harvard University

• 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

• 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 Settings

• 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: SettingsTypes 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: SettingsTranslating 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

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

Interpretation: Graphs Settings

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?

How Has Graphing Changed? Settings

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.

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.

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

Project: Differential Equations Settingsfrom 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.

SARS in Hong Kong: No quarantine Settings

Analyzed using 2003 World Health Organization data from Hong Kong

Analyzed Using 2003 World Health Organization data from Hong Kong

How Widespread are these Changes? SettingsExample: 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:

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

Challenges of Future Settings

• 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: Settings

• 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!