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

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

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### 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)
- 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 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

New problem types: Interpretation of the derivative Settings

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

Newer: Graphing with Parameters Settings

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

How Have Infinite Series Changed? Settings

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 Settings

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 Settings

(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

SARS in Hong Kong: With quarantine Settings

Analyzed Using 2003 World Health Organization data from Hong Kong

How Widespread are these Changes? SettingsExample: Calculus in US National Academy of Science study “Learning for Understanding” supported new syllabus. Made similar changes

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,

International IB Exam:

How Successful are These Changes? Settings

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!

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