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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)

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The teaching of mathematics what changes are on the horizon l.jpg

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

Deborah Hughes Hallett

University of Arizona

Harvard University


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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


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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


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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


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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”


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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


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New problem types: Interpretation of the derivative Settings

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


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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?


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How Has Graphing Changed? Settings

Previously, until early 1990s:

  • 50+ exercises to graph functions like

  • Occasional “proofs”: really calculations with answer given

  • No applications


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Newer: Graphing with Parameters Settings

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


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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.


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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.


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Newer: Application of Taylor series Settings

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


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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.


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SARS in Hong Kong: No quarantine Settings

Analyzed using 2003 World Health Organization data from Hong Kong


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SARS in Hong Kong: With quarantine Settings

Analyzed Using 2003 World Health Organization data from Hong Kong


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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:

  • Made similar changes


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    How Successful are These Changes? Settings

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


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    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??


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    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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